Loop Polynomial of Knots

Geometry & Topology 11 (2007) 1357–1475 1357

On the 2–loop polynomial of knots

TOMOTADA OHTSUKI

The 2–loop polynomial of a knot is a polynomial characterizing the 2–loop part of the Kontsevich invariant of the knot. An aim of this paper is to give a methodology to calculate the 2–loop polynomial. We introduce Gaussian diagrams to calculate the rational version of the Aarhus integral explicitly, which constructs the 2–loop polynomial, and we develop methodology of calculating Gaussian diagrams showing many basic formulas of them. As a consequence, we obtain an explicit presentation of the 2–loop polynomial for knots of genus 1 in terms of derivatives of the Jones polynomial of the knots. Corresponding to quantum and related invariants of 3–manifolds, we can formulate equivariant invariants of the infinite cyclic covers of knots complements. Among such equivariant invariants, we can regard the 2–loop polynomial of a knot as an “equivariant Casson invariant” of the infinite cyclic cover of the knot complement. As an aspect of an equivariant Casson invariant, we show that the 2–loop polynomial of a knot is presented by using finite type invariants of degree 3 of a spine of a Ä Seifert surface of the knot. By calculating this presentation concretely, we show that the degree of the 2–loop polynomial of a knot is bounded by twice the genus of the knot. This estimate of genus is effective, in particular, for knots with trivial Alexander polynomial, such as the Kinoshita–Terasaka knot and the Conway knot.

57M27; 57M25

Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday

1 Introduction

The Kontsevich invariant is a very strong invariant of knots, which dominates all quantum invariants and all Vassiliev invariants, and it is expected that the Kontsevich invariant classifies knots. A problem when we study the Kontsevich invariant is that it is difficult to calculate the Kontsevich invariant for any knot concretely. That is, the value of the Kontsevich invariant is presented by an infinite linear sum of Jacobi diagrams (a certain kind of uni-trivalent graphs), and it is not known so far how to calculate all terms of such a linear sum at the same time for an arbitrarily given knot.

Published: 23 July 2007 DOI: 10.2140/gt.2007.11.1357 1358 Tomotada Ohtsuki

Each term is a Vassiliev invariant, and there are algorithms to calculate it, but it is difficult to determine all terms at the same time. The infinite sum of the terms of the Kontsevich invariant with a fixed loop number (the first Betti number of uni-trivalent graphs) is presented by a polynomial [11; 21; 39]1; this presentation is called the “loop expansion”. In particular, it is known2 that the 1–loop part is presented by the Alexander polynomial. The polynomial presenting the 2–loop part is called the 2–loop polynomial. The 2–loop polynomial itself is a 2–variable polynomial invariant of knots. A table of the 2–loop polynomial for knots with up to 7 crossings is given by Rozansky [40]. The 2–loop polynomial of knots with the trivial Alexander polynomial can often been calculated by surgery formulas (Garoufalidis and Kricker[11], Kricker[19], Marche´[26]). The 2–loop polynomial for torus knots is explicitly presented by using a cabling formula (Marche´[25] Ohtsuki[35]). However, it is still difficult to obtain an explicit presentation of the 2–loop polynomial for an arbitrarily given knot, because the “language” to calculate the 2–loop polynomial has not been enough. An aim of this paper is to give a methodology to calculate the 2–loop polynomial for an arbitrarily given knot. We construct the 2–loop polynomial of a knot by calculating the rational version (Kricker[21]) of the Aarhus integral (Bar-Natan, Garoufalidis Rozansky and Thurston[2;3;4]) of a surgery presentation of the knot. The Lie algebra version of the Aarhus integral implies the perturbative expansion of a Gaussian integral, which is obtained by coupling the second-order part and higher-order part of the integral. In order to calculate the diagram version of the Aarhus integral explicitly, we introduce Gaussian diagrams, which present the second-order and higher-order parts of diagrams explicitly. We develop methodology to calculate Gaussian diagrams, showing many basic formulas of them. Corresponding to quantum and related invariants of 3–manifolds, we can formulate equivariant invariants of the infinite cyclic covers of knots complements (Section 1.3). Among such equivariant invariants, the 2–loop polynomial of a knot can be regarded as an “equivariant Casson invariant” of the infinite cyclic cover of the knot complement (Marche´[27] Ohtsuki[34]), while it is well known (see, eg, Lickorish[24]) that the Alexander polynomial can be regarded as an equivariant homology. One aspect of an equivariant Casson invariant is that a surgery formula for the 2–loop polynomial is given

1This was originally conjectured by Rozansky[39]. The existence of such rational presentations has been proved by Kricker[21] (though such a rational presentation itself is not necessarily a knot invariant in a general loop degree). Further, Garoufalidis and Kricker[11] deﬁned a knot invariant in any loop degree, from which such a rational presentation can be deduced. 2 This follows from the property called the Melvin–Morton–Rozansky conjecture (Bar-Natan and Garoufalidis[1]). See also[11; 21] and references therein.

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1359 by the equivariant linking number (Kojima and Yamasaki[17]) and equivariant finite type invariants of degree 3, while a surgery formula for the Alexander polynomial Ä is given by the equivariant linking number. Another aspect of an equivariant Casson invariant is that the 2–loop polynomial of a knot is presented by using finite type invariants of degree 3 of a spine of a Seifert surface of the knot (Theorem 4.4), Ä while the Alexander polynomial of a knot is presented by using finite type invariants of degree 1, ie, the Seifert form, of a spine of a Seifert surface of the knot.

By constructing the 2–loop polynomial using Gaussian diagrams along the latter aspect, in Theorem 4.7, we show the following estimate, which was conjectured by Rozansky [40], that

the degree of the 2–loop polynomial of a knot 2 the genus of the knot; Ä where the genus of a knot is the minimal genus of a Seifert surface of the knot. This implies that the non-zero coefﬁcients of the 2–loop polynomial of a knot lie in the hexagon whose edges are of length 2g for the genus g of the knot as shown in Table 1 and Table 2. This estimate is a reﬁnement of the estimate of the genus by the degree of the Alexander polynomial (see, eg,[24]), and, in particular, this estimate is effective for knots with trivial Alexander polynomial. For example, we see, in Example 4.13, that our bound is sharp for the Kinoshita–Terasaka knot and the Conway knot whose Alexander polynomial is trivial, while genera of them and many knots has been determined by Gabai[8] geometrically and by Ozsv ath´ and Szabo´[38; 37] using the knot Floer homology.

n 2 1 0 1 2 m 2 ı D m 1 ı ˇ ˇ ı D m 0 ˇ ˛ ˇ D m 1 ı ˇ ˇ ı D m 2 ı D n m Table 1: The non-zero coefﬁcients of t1 t2 in the 2–loop polynomial ‚K .t1; t2/ of a knot K of genus 1, where ˛; ˇ; ; ı are some integers given by Theorem 3.1 and Theorem 3.7.

Further, by calculating Gaussian diagrams, we show explicit presentations of the 2–loop polynomial for some knots. We show, in Theorem 3.7, that the 2–loop polynomial

Geometry & Topology, Volume 11 (2007) 1360 Tomotada Ohtsuki

n 6 5 4 3 2 1 0 1 2 3 4 5 6 m 6 3 3 3 3 3 3 3 D m 5 3 3 D m 4 3 2 2 2 2 2 3 D m 3 3 2 2 3 D m 2 3 2 1 1 1 2 3 D m 1 3 2 1 1 2 3 D m 0 3 2 1 1 2 3 D m 1 3 2 1 1 2 3 D m 2 3 2 1 1 1 2 3 D m 3 3 2 2 3 D m 4 3 2 2 2 2 2 3 D m 5 3 3 D m 6 3 3 3 3 3 3 3 D n m Table 2: The non-zero coefﬁcients of t1 t2 in the 2–loop polynomial ‚T .7;2/.t1; t2/ of the torus knot T .7; 2/ of type .7; 2/, whose genus is 3;

‚K .t1; t2/ of a knot K of genus 1 is presented by

1 2 1 1 Á ‚K .t1; t2/ V .1/ 3V .1/ d d .T2;1 T1;0/ d.d 1/T2;0 D 24 K000 C K00 C 3 2 1 2 1 2 7 1 Á V . 1/ .5d 5d 1/T1;0 d.5d 1/T2;0 5d d T2;1 ; 16 K0 C C 2 3 C 3 1 where VK .t/ denotes the Jones polynomial[15] of K, and d V .1/. Here, we D 6 K00 put Tn;m , for integers n; m with 0 2m n, by Ä Ä 1 n m m n n m m m n m m n n n m n (1) Tn;m t t t t t t t t t t t t D " 1 2 C 1 2 C 1 2 C 1 2 C 1 2 C 1 2 t nt m t mt n t m nt m t mt m n t n mt n t nt n m 12; C 1 2 C 1 2 C 1 2 C 1 2 C 1 2 C 1 2 where we put3 " 1 if 0 < 2m < n, and " 2 if 2m 0 or n; for example, D D D 2 2 1 1 1 2 2 1 T2;1 t t2 t1t t1t t t2 t t t t 6; D 1 C 2 C 2 C 1 C 1 2 C 1 2 3 2 3 2 3 3 2 1 2 T3;1 t t t t2 t t t1t t t t1t D 1 2 C 1 C 1 2 C 2 C 1 2 C 2 1 2 1 3 2 2 3 3 1 3 2 t t t t t t2 t t t t t t 12: C 1 2 C 1 2 C 1 C 1 2 C 1 2 C 1 2 Further, we show, in Proposition 2.4 and Proposition 2.5, that the 2–loop polynomial KT C for the Kinoshita–Terasaka knot Km and the Conway knot Km are presented by

‚ KT .t1; t2/ m .2T1;0 2T2;0 2T2;1 T3;1/; Km D C 3 n m That is, we set " 1; 2 so that the coefﬁcient of t t in Tn;m is equal to 1. D 1 2

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1361

‚ C .t1; t2/ m .2T1;0 2T2;0 2T2;1 T3;1/ Km D C 3 1 2m T1;0 T2;1 T3;1 T4;0 T4;2 T5;0 T5;1 T6;2 T6;3 : C C C C C 2 C This implies that the 2–loop polynomial is sensitive to mutation, unlike the Alexander and Jones polynomials. The paper is organized as follows. In Section 1, we review the deﬁnition of the 2– loop polynomial and its construction by the rational version of the Aarhus integral, introducing Gaussian diagrams. Further, we give a survey on equivariant invariants corresponding to quantum and related invariants. In Section 2, we give concrete presentation of the 2–loop polynomial for knots in terms of the Kontsevich invariant of their surgery presentation when the surgery is along knots, and calculate the 2–loop polynomial for the .4nm 1; 2n/ two-bridge knot, the Kinoshita–Terasaka knot, and C the Conway knot. In Section 3, we give explicit presentations of the 2–loop polynomial for knots of genus 1 in terms of derivatives of the Jones polynomial of them. In Section 4, we show the genus bound of the degree of the 2–loop polynomial, calculating the 2–loop polynomial for knots of any genus. Further, we show clasper surgery formulas for the 2–loop polynomial. In Section 5, we show many formulas of Gaussian diagrams used in the other sections, developing methodology of calculating them. See Figure 1 for relations among these sections. 2–loop polynomial of : Conway knot 2–loop polynomial of XX (Section 2.4) knots with .t/ 1 XX K XXz 2–loop polynomial of ¨¨* D Aarhus integral ¨ 2–loop polynomial of ¨¨* Kinoshita–Terasaka ¨ ¨ original version ¨ knots given by ¨ knot (Section 2.3) - surgery presentation rational version H ¢¢ - 2–loop polynomial of (Section 2.1 H¢ Section 3.4 ( H some two-bridge ( Section 5.2)) ¢ ( SectionHH 5.3) knots (Section 2.2) ( ¢ Hj Clasper surgery 2–loop polynomial of ¢ 2–loop polynomial of formula - knots given by ¢ Section 3.3 - knots of genus 1 spines of Seifert surfacesH (Section 3.1) (Section 3.2, Section 4.3) HH (Section 3.2, Section 4.1, H H Section 4.2) Hj genus bound by 2–loop polynomial (Section 4.2 ( Section 5.4)) Figure 1: Relations among sections (

Acknowledgments

The author would like to thank Andrew Kricker and Julien Marche´ for stimulating discussions and comments, and Kazuo Habiro, Stavros Garoufalidis, Taizo Kanenobu,

Geometry & Topology, Volume 11 (2007) 1362 Tomotada Ohtsuki

Hiroshi Goda, Mikami Hirasawa, Atsushi Ishii for helpful comments on the 2–loop polynomial and genera of knots. He would also like to thank the referee for careful reading of the manuscript.

1.1 Deﬁnition and properties of the 2–loop polynomial

In this section, we review the definition of the 2–loop polynomial. Further, we introduce the reduced 2–loop polynomial, and show a property of it. The Kontsevich invariant is defined in the space of Jacobi diagrams on S 1 , which we define as follows. For a 1–manifold X , a Jacobi diagram on X is the manifold X together with a uni-trivalent graph such that univalent vertices of the graph are distinct points on X and each trivalent vertex is vertex-oriented, where a vertex-oriented trivalent vertex is a trivalent vertex such that a cyclic order of the three edges around the trivalent vertex is fixed. In figures we draw X by thick lines and the uni-trivalent graphs by thin lines, in such a way that each trivalent vertex is vertex-oriented in the counterclockwise order. We define the degree of a Jacobi diagram to be half the number of univalent and trivalent vertices of the uni-trivalent graph of the Jacobi diagram. We denote by A.X / the quotient vector space spanned by Jacobi diagrams on X subject to the following relations, called the AS, IHX, and STU relations respectively,

; D

: D

The Kontsevich invariant Z.K/ [18] of a knot K is defined to be in A.S 1/; for details of its constructions, see, eg,[33]. The loop expansion of the Kontsevich invariant is defined in the space of open Jacobi diagrams. An open Jacobi diagram is a vertex-oriented uni-trivalent graph. We denote by A. / the quotient vector space spanned by open Jacobi diagrams subject to the AS and IHX relations. The Poincare–Birkhoff–Witt isomorphism (PBW isomorphism) A. / A. / is defined by W ! #

(2) D D ; 7!

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1363 for any diagram D, where the box denotes the symmetrizer,

1 Á (3) : D n! C C C n lines 2 3 A label of a power series f . / c0 c1 c2 c3 implies that „ D C „C „ C „ C f . / „ „ „ (4) c0 c1 „ c2 c3 „ : „ D C C C „ C

f . / f . / Note that „ „ D by the AS relation, in the notation of this paper. Any open Jacobi diagram can be presented by a trivalent graph with labels on its edges. It is known[11; 21; 39] that the Kontsevich invariant of a knot K has a presentation,

1 sinh. =2/ 1 pi;1.e„/=K.e„/ 2 log =„2 2 log K.e„/ „ ﬁnite 1 X pi;2.e„/=K.e„/ log Z.K/ t D C i pi;3.e„/=K.e„/

terms of 3–loop, 4–loop, presented in the same way; C where log denotes the logarithm with respect to the disjoint-union product of open Ja- t cobi diagrams, K .t/ denotes the Alexander polynomial, and pi;j .e„/ is a polynomial in e˙„ . This presentation is called the loop expansion. We denote its 2–loop part by

pi;1.e„/=K.e„/ ﬁnite X p .e /= .e / Z.2–loop/.K/ i;2 „ K „ : D i pi;3.e„/=K.e„/

The 2–loop part is characterized by the polynomial,

X 1 1 1 pi;1.t1/pi;2.t2/pi;3.t3/ ޑŒt ; t ; t =.t1t2t3 1/; 2 1˙ 2˙ 3˙ D i where the equivalence “ ” is generated by " " " (5) f .t1; t2; t3/ f .t ; t ; t / .1/ .2/ .3/

Geometry & Topology, Volume 11 (2007) 1364 Tomotada Ohtsuki for any " 1 and any permutation on 1; 2; 3 , which is derived from the symmetry D ˙ f g of the graph, while the relation t1t2t3 1 is derived from the IHX relation. The D symmetrization of the above polynomial with respect to the symmetry of the graph is given by

X " " " 1 1 1 ‚K .t1; t2; t3/ pi;1.t /pi;2.t /pi;3.t / ޑŒt ; t ; t =.t1t2t3 1/; D .1/ .2/ .3/ 2 1˙ 2˙ 3˙ D " 1 D˙ 1 1 where runs all permutations on 1; 2; 3 . Putting t3 t t , we denote it by f g D 1 2 ‚K .t1; t2/, and call it the 2–loop polynomial of K, (Note that this normalization of ‚K .t1; t2/ is 12 times the normalization in[40].) We also denote the 2–loop polynomial by ‚.K/.

The 2–loop polynomial of the mirror image K of a knot K satisfies that ‚K .t1; t2/ x x D ‚K .t1; t2/, since Z.K/ is obtained from Z.K/ by changing the sign of the part of x odd degree. 1 A particular value ‚K .t; 1/ is a symmetric polynomial in t ˙ divisible by t 1 (since 2 ‚K .1; 1/ 0 by definition) and hence, divisible by .t 1/ . As in[35], we define the D reduced 2–loop polynomial by

‚K .t; 1/ 1 ‚K .t/ ޑŒt ˙ ; y D .t 1=2 t 1=2/2 2 1 which is a symmetric polynomial in t ˙ . As shown in[35], this presents the sl2 reduction of the 2–loop polynomial.

We obtain the formulas of the following proposition for ‚K .1/ and ‚K . 1/ by y y rewriting formulas in[35] and in[10; 28] respectively. Recall that the Jones polynomial 1=2 VL.t/ ޚŒt (which we also denote by V .L/) of an oriented link L is deﬁned by 2 ˙ the skein relation Á Á Á t V t 1V .t 1=2 t 1=2/ V D and the normalization V .the trivial knot/ 1. Here, the three pictures in the formula D imply three oriented links, which are identical except for a ball, where they differ as shown in the pictures.

Proposition 1.1 (See[35] and[10; 28]) The reduced 2–loop polynomial of a knot K at t 1 satisﬁes that D ˙ 1 1 ‚K .1/ 2 v3.K/ V .1/ V .1/; y D D 18 K000 C 6 K00 1 ‚K . 1/ V . 1/VK . 1/; y D 12 K0

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1365

1 where v3.K/ denotes 4 times the coefﬁcient of the diagram in log Z.K/, which is an integer-valued primitive Vassiliev invariant of degree 3. t

Proof We obtain the ﬁrst formula of the proposition, as follows. It is shown in[35] that ‚ .1/ is equal to 2v .K/. In particular, it is a primitive Vassiliev invariant of y K 3 degree 3, which is unique up to a scalar multiple. Hence, it is equal to a scalar multiple 4 3 h of the coefﬁcient of h in the expansion of VK .e /. We can determine the scalar of the multiple by calculating an example, say, see Example 3.6. It follows that

1 3 h ‚K .1/ the coefficient of h in VK .e / : y D 3 h d 3 h h 3h Further, since the third derivative of VK .e / is given by dh3 VK .e / VK000.e /e h 2h h h 3 h D C 3V .e /e V .e /e , the coefficient of h =6 in VK .e / is equal to V .1/ K00 C K0 K000 C 3VK00 .1/. Therefore, we obtain the first formula of the proposition. We obtain the second formula of the proposition, as follows. The Casson–Walker invariant of the double branched cover of S 3 branched along K is presented in[28] (see also[9]) by a linear sum of the signature of K and V . 1/=VK . 1/. On the K0 other hand, it is also presented in[10] by a linear sum of the signature of K and 2 ‚K .1; 1/ ‚K . 1; 1/ ‚K . 1; 1/ =K . 1/ , noting that ‚K .1; 1/ 0. This C C 2 D is further equal to 12‚. 1/=K . 1/ from the definition of the reduced 2–loop y polynomial. Hence, ‚. 1/ is equal to a scalar multiple of V . 1/VK . 1/, noting y K0 that VK . 1/ K . 1/. We can determine the scalar of the multiple by calculating an D example, say, see Example 3.6, and this gives the second formula of the proposition.

1.2 Construction of the 2–loop polynomial

In this section, we review a construction of the 2–loop polynomial by the rational version of the Aarhus integral. We introduce Gaussian diagrams to calculate the rational version of the Aarhus integral explicitly. Along this construction, we calculate the 2–loop polynomial in Sections 2–4.

Consider a framed link K L, such that K is isotopic to the trivial knot with 0 [ framing, and the linking number of K and each component of L is equal to 0, and the 3–manifold obtained from S 3 by surgery along L is homeomorphic to S 3 . We denote by KL the knot obtained from K by surgery along L. The link K L is called a [

4 Putting V .eh/ 1 c .K/h2 c .K/h3 , we can show that c is primitive, ie, c is additive K D C 2 C 3 C 3 3 with respect to the connected sum of knots, from the fact that VK .t/ is multiplicative with respect to the connected sum of knots.

Geometry & Topology, Volume 11 (2007) 1366 Tomotada Ohtsuki

surgery presentation of the knot KL . For example,

(6) KL ; K L D [ D D

where we depict K and KL by thick lines, and depict L by thin lines. We explain how to calculate the 2–loop polynomial of KL from the Kontsevich invariant of K L. [ We review how to calculate the loop expansion of the Kontsevich invariant of KL from the Kontsevich invariant of K L, following an idea of Kricker[21], assuming, for [ simplicity, that L is a knot; for details see[11; 21]. The Kontsevich invariant Z.K L/ [ of K L is deﬁned to be in A.S 1 S 1/. We label the two components of S 1 S 1 by [ t t and x respectively. A partially open Jacobi diagram on S 1 , where and S 1 are „ t labeled by and x respectively, is a vertex-oriented uni-trivalent graph such that some „ of the univalent vertices of the graph are labeled by and the other univalent vertices „ are distinct points on S 1 . The PBW isomorphism A. S 1/ A. S 1/ is „W t ! # t deﬁned by applying the map (2) to the univalent vertices labeled by . We identify „ A. S 1/ and A.S 1 S 1/ by the isomorphism A. S 1/ A.S 1 S 1/, which is # t t # t ! t obtained by closing the two end points of . It is shown that 1Z.K L/ A. S 1/ # [ 2 t is equal to a linear sum of Jacobi diagrams whose –labeled„ vertices are given by „ labels of polynomials in e˙„ , by using the formula[6],

e 1 Á „ Z ; „ „ D

2 3 where a label of a power series f . / c0 c1 c2 c3 implies that „ D C „C „ C „ C

f . / „ „ „ c c c c „ : 0 1 „ 2 „ 3 D C C C „ C

An open Jacobi diagram on , where the two ’s of are labeled by and x t t „ respectively, is a vertex-oriented uni-trivalent graph each of whose univalent vertices is labeled by either or x. We denote by A. / the quotient vector space spanned by „ t Jacobi diagrams on subject to the AS and IHX relations. The PBW isomorphism t A. / A. / is deﬁned by applying the map (2) to the –labeled vertices W t ! # t # „ and the x–labeled vertices respectively. This isomorphism is equal to the composition 1 of and x A. / A. /. We choose a pre-image of Z.K L/ in „ W t ! t # „ [

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1367

A. /, which we denote by 1Z.K L/, by choosing a 1–tangle K L whose t [ [ L closure is isotopic to K L. The Kontsevich invariant of the surgery along L is [ described by the rational version[21] of the Aarhus integral[2;3;4], as follows, Z 1 Z.K L/dx 1 L [ (7) Z.KL/ Z ; D 1 Z.U /dx L ˙ where U denotes the trivial knot with 1 framing whose sign we choose depending ˙ ˙ on the sign of the framing of L, and, by deﬁnition (see[23]), Z.K L/ is obtained L [ from Z.K L/ by connect-summing to the L–component of Z.K L/. An idea [ [ of Kricker[21] is to calculate the loop expansion of the Kontsevich invariant of KL from (7).

When we calculate the Aarhus integral, it is often convenient to use the link relation “ ”[2;3;4], which is deﬁned by .Á~/

D 0 .Á~/ for any diagram D, where we put

: D C C C C The link relations is a relation which relates pre-images in A. / of an element in t A. / by the PBW map x , and it is known[2;3;4] that the result of the Aarhus t # integral does not depend on the difference derived from the link relation.

To calculate the rational version of the Aarhus integral explicitly, we use Gaussian diagrams, which we introduce as follows. We denote by a chord written in a double line an exponential chord; for example,

f f f f 1 1 f (8) f A.two intervals/; D C C 2 C 6 f C 2 f f Á exp A. /; D t 2 where exp denotes the exponential with respect to the disjoint-union product of open Jacobi diagrams.t We call a Jacobi diagram with exponential chords a Gaussian diagram. Further, we denote by a uni-trivalent graph written in double lines its exponential; for

Geometry & Topology, Volume 11 (2007) 1368 Tomotada Ohtsuki example,

1/24 1 Á ˆ exp ; Á Á 24

1/48 1 Á 1 ; Á Á C 48 where is the element in A. / whose closure in A.S 1/ is equal to the Kontsevich # invariant of the trivial knot, and ˆ is the element called an associator which is the Kontsevich invariant of an elementary q–tangle; they are basic elements appearing in a combinatorial construction of the Kontsevich invariant; for details, see, eg,[33]. Here, in the above formulas (and throughout this paper), we write ˛ ˇ (resp. ˛ ˇ) if Á Á.2/ ˛ ˇ is a linear sum of Jacobi diagrams with at least 3 (resp. 2) trivalent vertices, where we do not count trivalent vertices generated by attached power series. If a uni-trivalent graph has 2 trivalent vertices, we can put it in any way in a Jacobi diagram modulo the equivalence; for example,

Á since their difference equals 0: Á

So, we write the previous diagrams as .

We explain how to calculate the rational version of the Aarhus integral in (7) concretely, using Gaussian diagrams. As mentioned above, 1Z.K L/ is presented by a linear [ sum of Jacobi diagrams whose –labeled vertices„ are given by labels of polynomials „ in e˙„ . For simplicity, as in Section 2.1 and Section 2.2, we assume that it is presented by

1 f .t/ Z.K L/ .1 ˇ/ „ [ Á C for some polynomial f .t/ in t 1 putting t e and some ˇ which is a linear sum of ˙ D „ Jacobi diagrams each of which has at least two trivalent vertices; recall that a double

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1369 line is deﬁned in (8) and a label of a power series of is deﬁned in (4), that is, „

1 1 f .t/ f .t/ f .t/ f .t/ f .t/ f .t/ f .t/ ; D C C 2 C 6 C

f .t/ „ „ c c c c „ ; 0 1 „ 2 „ 3 D C C C „ C

2 3 where f .e / c0 c1 c2 c3 . By deﬁnition (see[23]), Z.K L/ is „ D C „C „ C „ C L [ obtained from Z.K L/ by connect-summing to the L–component of Z.K L/, [ [ that is,

Â Ã 1 f .t/ 1 Z.K L/ 1 ˇ : „ L [ Á C C 48

1 By calculating of a Gaussian diagram (for details see Section 2.1), we have that

.t/=2 1Z.K L/ .1 ˇ / L [ Á C 0 for some polynomial .t/ in t 1 with .t/ .t 1/, which gives the Alexander ˙ D polynomial of KL , and for some ˇ0 which is a linear sum of Jacobi diagrams each of which has at least one trivalent vertex. As in[11; 21], the rational version of the Aarhus integral is deﬁned by

Z 1Z.K L/ dx ; ˇ L [ Á 0 1=2.t/ where the bracket D1; D2 is deﬁned to be the sum of the diagrams obtained by h i connecting the univalent vertices of D1 and the univalent vertices of D2 if D1 and D2 have the same number of univalent vertices, and 0 otherwise. Hence, by (7),

Â Ã 1 1 (9) Z.KL/ 1 ; ˇ ; Á ˙ 16 0 1=2.t/

Geometry & Topology, Volume 11 (2007) 1370 Tomotada Ohtsuki where we choose the same sign in the formula as the sign of the framing of L, since the normalization factor in (7) is calculated as follows:

Z Z Â Â ÃÃ 1 1 1=2 1 Z U dx ˙ 1 dx L ˙ Á C 24

1=2 Z Â ˙ Â 1 ÃÃ 1 1 1 dx ; 1 1 : Á C 16 D 1=2 C 16 Á 16

From (9), we obtain an explicit presentation of the 2–loop polynomial of KL by calculating ˇ0 concretely.

1.3 Equivariant invariants of the inﬁnite cyclic covers of knot complements

The 2–loop polynomial of a knot can be regarded as an “equivariant Casson invariant” of the inﬁnite cyclic cover of the knot complement[27; 34]. The aim of this section is to see this from the viewpoint of quantum and related invariants of 3–manifolds and equivariant invariants corresponding to them. We give a survey on quantum and related invariants of 3–manifolds, and explain how Casson invariant behaves among them. Further, we see what are equivariant invariants corresponding to them, and consider relations of the 2–loop polynomial to these equivariant invariants, which would be meaningful for future directions of the study of these invariants.

We review quantum and related invariants of 3–manifolds; for details, see eg[33]. Let M be a closed 3–manifold, and let L be a framed link in S 3 such that M is 3 obtained from S by surgery along L. For simplicity, we consider the sl2 case. Let Vn be the n–dimensional irreducible representation of the quantum group Uq.sl2/, whose quantum dimension is Œn .qn=2 q n=2/=.q1=2 q 1=2/, and let r be an D odd integer 3, and put exp.2p 1=r/. Then, the quantum SO.3/ invariant D of M is deﬁned by

SO.3/ X ˇ .M / .normalization constant/ Œn1 Œnl Q.L Vn ; ; Vn /ˇ ރ; r D I 1 l q 2 n1; ;nl D where the sum of each ni runs over ni 1; 3; ; r 2, and Q.L Vn ; ; Vn / de- D I 1 l notes the quantum invariant of L whose components are associated with Vn1 ; ; Vnl , 1 which is a polynomial in q˙ . Further, for simplicity, let M be an integral homology 3–sphere. Then, the perturbative SO.3/ invariant SO.3/.M / is deﬁned to be an

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1371

SO.3/ arithmetic limit of r .M / as r , ie, ! 1

SO.3/ X1 n .M / n.q 1/ ޚŒŒq 1 D 2 n 0 D where n ’s are uniquely characterized by

.r 3/=2 SO.3/ X n .r 1/=2 .M / n. 1/ modulo . 1/ in ޚŒ r Á n 0 D for any prime r 5. Further, as shown in[13; 14], the perturbative invariant has an expansion of the form

SO.3/ X1 2 n .M / an.q 1/.q 1/ .q 1/ D n 0 D for an ޚŒq, and each quantum invariant is derived from the perturbative invariant by 2 substituting q in this expansion, ie, D SO.3/ SO.3/ ˇ r .M / .M /ˇq ; D D where the substitution is taken in the above expansion. Each perturbative invariant is derived from the LMO invariant through the weight system (see[33]), and, in this sense, the LMO invariant is universal among all perturbative invariants. Further, as shown in[22], the LMO invariant of integral homology 3–spheres is universal among all ﬁnite type invariants; in particular, the degree d part of the LMO invariant are of ﬁnite type of degree d . See Figure 2 for relations among these invariants.

Casson invariant .M / of an integral homology 3–sphere M appears in the degree 1 part of these invariants as follows; for details, see eg[33]. The degree 1 part of the LMO invariant is given by Casson invariant, and so is the ﬁnite type invariant of degree 1. In particular, Casson invariant has a clasper surgery formula,

Â Ã Â Ã (10) 1; D where the ﬁrst picture implies a surgery on an integral homology 3–sphere along a graph clasper of the form of a graph (see Section 4.3, for the deﬁnition of a graph clasper). Further,

(11) SO.3/.M / 1 6 .M /.q 1/ .higher terms/; D C C

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Quantum invariants

6 r q e2p 1=r ! 1 D ? coefﬁcients are of ﬁnite type - Perturbative invariants Finite type invariants degree 0 invariant: the order of H1 @I degree 1 invariant: Casson invariant universal @ universal @ @

LMO invariant

Figure 2: Quantum and related invariants of 3–manifolds and hence, (12) SO.3/.M / 1 6 .M / . 1/ modulo . 1/2 in ޚŒ r Á C for any prime r 5. In particular, .M / modulo r ޚ=rޚ is determined by SO.3/ 2 r .M / for any prime r 5, as shown in[29; 30]. Corresponding to the invariants shown in Figure 2, we consider equivariant invariants of the infinite cyclic covers of knot complements, as shown in Figure 3. Corresponding to a quantum invariant of 3–manifolds defined from a modular category Vm Vi i I , an invariant T .K/ of a knot K is defined as an equivariant version of f g 2 the quantum invariant for the infinite cyclic cover of the complement of K, as in [36]. Roughly speaking, T Vm .K/ is defined to be the characteristic polynomial of the quantum invariant of a 3–cobordism obtained from S 3 K by cutting it along a Seifert surface whose boundary is associated with Vm . To be precise, when a link Vm K L is a surgery presentation of a knot KL as in (6), T .KL/ is defined to be the [ characteristic polynomial of a matrix whose entries are quantum invariants of a tangle L, where L is a tangle obtained from L by cutting along a disk bounded by K; for y y details, see[36]. For example, for K L shown in (6): [

L y D

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Corresponding to finite type invariants of 3–manifolds, loop finite type invariants of pairs of integral homology 3–spheres and knots are defined as follows. Consider pairs .M; K/ such that M is an integral homology 3–sphere and K is an oriented knot in M , and consider a move between two pairs .M; K/ and .M 0; K0/ such that .M 0; K0/ is obtained from .M; K/ by surgery on a Y graph embedded in M K whose leaves have linking number zero with K, where a Y graph is a graph clasper of the form:

Loop finite type invariants of such pairs are defined similarly as a definition of Vassiliev invariants using this move instead of a crossing change; for details, see[11] 5 (see also [32]). The weight systems of loop finite type invariants are based on the grading of Jacobi diagrams given by the loop-degree, where the loop-degree of a Jacobi diagram on S 1 is defined to be half of the number given by the number of trivalent vertices minus the number of univalent vertices of the uni-trivalent graph of the Jacobi diagram, ie, an n–loop diagram is a diagram of loop-degree n 1. The loop expansion of the Kontsevich invariant (the rational Z invariant) is universal among loop finite type invariants[11]. Further, the sl2 reduction of the loop expansion of the Kontsevich invariant gives the loop expansion of the colored Jones polynomials. See Figure 3 for relations among these invariants, corresponding to relations shown in Figure 2. From the viewpoint that the 2–loop polynomial is an equivariant Casson invariant, we can expect some relations between the 2–loop polynomial and invariants shown in Figure 3, corresponding to the relations between Casson invariant and invariants in Figure 2 mentioned before. Corresponding to the clasper surgery formula (10), the 2–loop part of the Kontsevich invariant has a clasper surgery formula 0 1 t n B C Â Ã .2–loop/ B C .2–loop/ Z B C Z @ A D t m as shown in[11], where the left picture implies a surgery on a knot along a graph clasper of the form of a graph whose upper and lower loops have linking numbers n

5 The deﬁnition of loop ﬁnite type invariants also appears in the September 1999 version of[20].

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Equivariant quantum invariants

p p p p Loop expansionp of p the colored Jones polynomial Loop ﬁnite type invariants and quantum invariants pppppppppppppp loop-degree 0 invariant: Alexander polynomial @I loop-degree 1 invariant: @ universal @ universal 2–loop polynomial @ The rational Z invariant

Figure 3: Equivariant invariants of the inﬁnite cyclic covers of knot complements corresponding to the invariants shown in Figure 2

and m with the knot. Further, the 2–loop polynomial has other clasper surgery formulas shown in Section 4.3. Corresponding to the relation (11) between Casson invariant and the perturbative invariant, we have a relation between the reduced 2–loop polynomial and the 2–loop part of the loop expansion of the colored Jones polynomials as in[35]. Corresponding to the relation (12) between Casson invariant and the quantum invariant, we can expect that there would be some relations between the 2–loop polynomial and equivariant quantum invariants, though such relations are not formulated yet so far. Thus, the 2–loop polynomial would have a central role in the study of these invariants.

2 The 2–loop polynomial calculated from surgery presentations

In this section, we calculate the 2–loop polynomial of knots from their surgery presentations. We give concrete presentations of the 2–loop polynomial for knots in terms of the Kontsevich invariant of their surgery presentations of surgery along knots in Section 2.1. Further, we calculate the 2–loop polynomial for the .4nm 1; 2n/ two-bridge knot, C the Kinoshita–Terasaka knot, and the Conway knot in Section 2.2, Section 2.3, and Section 2.4, respectively.

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2.1 Surgery presentations of surgery along knots

In this section, we give a concrete presentation of the 2–loop polynomial of a knot in terms of the Kontsevich invariant of a surgery presentation of the knot whose surgery is along a knot.

Consider a link K L, such that K is isotopic to the trivial knot with 0 framing, L is [ a knot with 1=m framing, and the linking number of K and L is equal to 0. We ˙ denote by KL the knot obtained from K by surgery along L. The link K L is a [ surgery presentation of the knot KL . The aim of this section is to give a concrete presentation of the 2–loop polynomial of KL in terms of the Kontsevich invariant of K L. [ An idea of[5] to compute a rational surgery is to deﬁne the Kontsevich invariant of a string with a rational framing to be the Kontsevich invariant of a Hopf chain, for example, as follows,

Â 1=m Ã Â Ã 1 Z Â z Ã framing˙ 1 1 Z 1 z Z m dz 16 L framing y D ˙ y ! 1=m2 1 .m 1/.m 2/ 1 : 1=2m 48 48m Á y ˙ C ˙ For detailed and general formulas, see[5]. Hence, by putting the Kontsevich invariant of a string of a rational framing in this way, we can also apply the rational version of the Aarhus integral.

1 Let us calculate the 2–loop polynomial of KL in a simple case that Z.K L/ is [ given by „

f Â 1 .m 1/.m 2/ Ã 1Z.K L/ 1 ; 2 „ [ Á C 48m ˙ 48m for some polynomial f .e / in e ; e , satisfying that f .1/ 1=2m, since L has „ „ „ D ˙ the framing 1=m. The reason why we add the last factor in the above formula is ˙ 1 that this term is natural in the sense that, if f was a scalar in 2 ޚ, the above formula presents, modulo “ ”, the Kontsevich invariant of the trivial knot with 2f framing. Á From the deﬁnition of Z, L Â 2 Ã 1 f 1=m 1 .m 1/.m 2/ Z.K L/ 1 C : „ L [ Á C 48 ˙ 48m

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Further, by Lemma 5.13,

Â 1 1 1 Ã 1 f x 1 ; .Á~/ C 12 4 C 12 where we introduce two markings by

1 f 1 f 1 1 (13) ; : D 2 C 2 D 2 2

1 Hence, Z.K L/ 1 ˇ ; L [ .Á~/ C where 1=m2 1 1 1 1 .m 1/.m 2/ ˇ C : D 48 C 12 4 C 12 ˙ 48m

The rational version of the Aarhus integral is calculated as follows, Z 1Z.K L/dx ; ˇ ; L [ D Â Ã where we put exp : D t

Here, we deﬁne the marking of a circle by

1 2.f f / (14) C D for f and its conjugate f deﬁned by f .e / f .e /. In particular, we have that „ D „ 1 : D 4

For the diagram of ˇ, we have that

; 2 ; D

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; 2 ; D ; 2 ; D

1 Â Ã ; ; 4 D 2 D C C

8 f .e0/ 4 4 4 ; D C C C C where f .e0/ 1=2m. Hence, D ˙ Z 1=m2 1 1 1 1 1Z.K L/dx 1 C L [ Á C 24 C 6 ˙ 3m 12 1 1 1 1 .m 1/.m 2/ : 6 C 3 C 3 C 12 ˙ 48m

Further, as we explained in Section 1.2, we have, from (9), that

2 2 .2–loop/ m 2 1=m 1 1 1 (15) Z .KL/ C C D ˙ 48m C 24 C 6 ˙ 3m 1 1 1 1 1 : 12 6 C 3 C 3 C 12

Therefore, from the deﬁnition of the 2–loop polynomial, we obtain the following proposition, by putting .t/ m f .t/ f .t 1/ and ı.t/ f .t/ f .t 1/. D ˙ C D

Proposition 2.1 Let K L and KL be as above, satisfying that [ Â Ã 1 .t/=2m ı.t/=2 1 .m 1/.m 2/ Z.K L/ ˙ C 1 2 „ [ Á C 48m ˙ 48m

1 1 for polynomials .t/ and ı.t/ in t; t satisfying that .1/ 1, .t / .t/ 1 D D and ı.t / ı.t/. Then, the 2–loop polynomial of KL is presented, modulo the D

Geometry & Topology, Volume 11 (2007) 1378 Tomotada Ohtsuki equivalence (5), by

1 2 2 (16) ‚K .t1; t2/ .t1/.t2/ .m 2/.t3/ .t1/.t2/ m 1 L ˙ 4m C m ı.t1/ı.t2/ 1 1 2 .t3/ .t1/.t2/ .t1/.t3/ .t3/ 4 C 2 C C 2 2 m ı.t1/ .t2/ .t2/ .t3/ ; ˙ 8 and its reduced 2–loop polynomial is presented by

2 1 3 2 2 m ı.t/ 2 ‚K .t/ .t/ .t/ .2m 3/.t/ 2m 2 .t/ 2 : y L D 12m C C C ˙ 12 C

1 Remark 2.2 More generally, if Z.K L/ is given by the following form, „ [ Â Ã 1 .t/=2m ı.t/=2 1 .m 1/.m 2/ Z.K L/ ˙ C 1 ˇ ; 2 „ [ Á C 48m ˙ 48m C where ˇ is a linear sum of Jacobi diagrams each of which has two trivalent vertices, .2–loop/ then we can show in a similar way as above that Z .KL/ is presented by .2–loop/ Z .KL/ the right-hand side of (15) ; ˇ ; D C and the 2–loop polynomial ‚KL .t1; t2/ is given by the sum of the right-hand side of (16) and the polynomial corresponding to the last term of the above formula.

2.2 The 2–loop polynomial of the .4nm 1; 2n/ two-bridge knot C In this section, we calculate the 2–loop polynomial of the .4nm 1; 2n/ two-bridge C knot as an application of Proposition 2.1 (and Remark 2.2).

The .4nm 1; 2n/ two-bridge knot is the knot given by C

−m ; n where we mean k full twists by a boxed “k ”; for example,

3 : D

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The above knot is obtained from the following link K L by 1=m surgery along the [ component L labeled by x:

1=m framing K L [ D n

Let K L0 be K L putting the framing of L to be 0. Then, [ [ x

x n x n 0 framing K L0 closure of : [ D D D n

Hence, 2n

ν2 ν1/2

S2 Φ −1/24

− S2 exp n n 1/24 Â 1 Ã 1 Φ −1 Z.K L0/ cl. of S2 cl. of 1 : − L [ D − Á 1 C 16 „ t 1 t t t −1/24

S2 Φ n

ν1/2 exp n

Since t t Â t t Ã −nt Â Ã −nt −nt 1 n t 1 n ; Á Á we have, by Lemma 5.8, that

1 Z.K L/ „ [

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−n t Â 1 n t t .m 1/.m 2/ Ã 1 Á n +1/2m C 48m2 C 24 C 48m

Â 1 .m 1/.m 2/ Ã − nt+n +1/2m 1 ˇ ; Á C 48m2 C 48m C where 2n 1=m 2n 1=m 2n 1=m n ˇ C C C ; D 4 C 4 3 C 24 under the notation (13), putting f nt n 1 . D C C 2m We apply Proposition 2.1 (and Remark 2.2) to the above formula of 1Z.K L/, [ putting .t/ 1 nm.t t 1 2/ and ı.t/ n.t t 1/, to obtain „ D C D 1 2 2 (17) ‚K .t1; t2/ .t1/.t2/ .m 2/.t3/ .t1/.t2/ m 1 L 4m C m ı.t1/ı.t2/ 1 1 2 .t3/ .t1/.t2/ .t1/.t3/ .t3/ 4 C 2 C C 2 2 m ı.t1/ .t2/ .t2/ .t3/ C 8 where the additional part is the polynomial corresponding to ; ˇ :

Here, the marking of a circle is deﬁned in (14). We calculate this for each diagram of ˇ as ; 2 ; D

1 ; ; D 2 1 ; ; D 4 t ; 4 8 4 t : D t C

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1381

2n 1=m 2n 1=m Hence ; ˇ C C D 24 6 nÂ t Ã 2 t : C 6 t C

Since the markings of a crossing and a circle stand for the labels of .t/=2m and m=2.t/ respectively, from the deﬁnition of the 2–loop polynomial, the additional part of (17) is presented, modulo the equivalence (5), by

2 2n 1=m nm 1 C .t1/.t2/ .t1/ .t3/ .t1/ 1 2t2t t1 : 2 C 2 3 C Therefore,

1 2 2 ‚K .t1; t2/ .t1/.t2/ .m 2/.t3/ .t1/.t2/ m 1 L 4m C m ı.t1/ı.t2/ 1 1 2 .t3/ .t1/.t2/ .t1/.t3/ .t3/ 4 C 2 C C 2 2 m ı.t1/ .t2/ .t2/ .t3/ C 8 2 2n 1=m nm 1 C .t1/.t2/ .t1/ .t3/ .t1/ 1 2t2t t1 : C 2 C 2 3 C By substituting .t/ 1 nm.t t 1 2/ and ı.t/ n.t t 1/ and by symmetrizing D C D the formula, we obtain the following proposition.

Proposition 2.3 The 2–loop polynomial of the .4nm 1; 2n/ two-bridge knot is C presented by

! −m nm.n m/ nm.nm 1/ ‚ nm.nm 1/T1;0 T2;0 D 2 C C 2 n 3n2m2 nm 1 Á T2;1 ; 3 where Tn;m ’s are deﬁned in (1), and its reduced 2–loop polynomial is presented by 0 1 nm.n m/ ‚ B C 6 .2nm 1/.t t 1 2/: y @ A D 6 C

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2.3 The 2–loop polynomial of the Kinoshita–Terasaka knot

In this section, we calculate the 2–loop polynomial for the Kinoshita–Terasaka knot, from a surgery presentation of the knot. Unlike the case of the previous sections, we need, not the rational version, but the original version of the Aarhus integral to calculate the surgery, since the equivariant linking number of the surgery presentation is a scalar in this case.

KT The Kinoshita–Terasaka knot Km [16] is the knot given by

KKT : m D m

KT KT The 2–loop polynomial ‚.Km / of Km is a polynomial in m, as we see later. Note, in advance, that this polynomial consists only of terms of odd power of m, since KT KT KT KT KT ‚.K m/ ‚.Km / ‚.Km /, where Km denotes the mirror image of Km . D D

KT The Kinoshita–Terasaka knot K is obtained from the following link K L0 by m [ 1=m surgery along the middle component L0 ,

K L0 [ D D D

; D D

where in these pictures we mean by a thick line 2 parallel copies of the line.

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We calculate the Kontsevich invariant of K L0 comparing it to the Kontsevich [ invariant of the following link,

K L ; [ 0 D

which is isotopic to the trivial 2–component link. Since

0 1 −1 B C Â x x Ã B C 1 1 Z B C 1/24 1 ; B C Á C 96 C 48 @ A 1/24 x y

x y x y

x y (18) −1 t x y t t /24 x y x Â Ã Â 1 1 1 Ã 1Z t −1 1 ; Á C 96 C 96 48 „ t /24 x y y

the Kontsevich invariant of K L0 is presented by [

1 Z.K L0/ „ [ − t t t t −1 t /24 t 1 t 1 t /24 Â Ã t /24 1 1 t /24 1 : Á t /24 C C 32 C 24 t 1 t.t 1/ t −1 t /24

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Hence, the difference between the Kontsevich invariant of K L0 and K L is [ [ 0 given by

1 1 Z.K L0/ Z.K L0/ „ [ „ [ − t t − t t 1 t t −1

t /24 t /24 t /24 t /24 t /24 t /24 t /24 t /24 : Á t /24 t /24 t −1 t −1 t /24 t /24

By Lemma 5.3, the ﬁrst diagram is equivalent, modulo diagrams of the form , to

−t −1 t t t −1 t /24 t 1 t /24 Â Ã t /24 1 t /24 1 t /24 C 2 Á − t.t 1/ . / t 1 t /24

t t −1 t /24 t 1 t.t 1/ t 1 t /24 Â t t 1 Ã t /24 1 t 1 t t 1 t /24 1 C ; t /24 C 2 C 4 C − t.t 1/ t 1 t 1 t 1 t /24

where we obtain the equivalence by Lemma 5.9, noting that diagrams of the form KT 2 KT contribute to ‚.Km / by terms of m , though such terms vanish in ‚.Km / as we mentioned before. It follows from the above formulas that t.t 1/ t 1 Â 1 1 Ã t 1 t t 1 1 C 2 C t t 1 Z.K L0/ Z.K L / : [ [ 0 Á C 4 C „ „ . / t 1 t 1

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Since K L is isotopic to the trivial link, we have that [ 0 t.t 1/ t 1 1 1 1 t 1 t t Á 1 C 2 C t t 1 Z.K L0/ 1 : [ Á C 48 C 4 C „ . / t 1 t 1

Let L be L0 with 1=m framing. Then, similarly as in Section 2.1, 2 − 1=m 1 .m 1/.m 2/ 1Z.K L/ 1/2m 1 L C „ [ .Á/ C 48 48m t.t 1/ t 1 1 1 t 2 1 t t 1 Á C C t t : C 4 C t 1 t 1 Further, from the deﬁnition of , 1=2m 2=m2 1 .m 1/.m 2/ 1Z.K L/ 1 L C [ .Á/ C 48 48m t.t 1/ t 1 1 1 t 2 1 t t 1Á C C t t : C 4 C t 1 t 1 KT The 2–loop part of the Kontsevich invariant of Km is given by the Aarhus integral, as follows, t.t 1/ t 1 t 1 t t 1 .2–loop/ KT 2 1 1 Z .K / ; C C t t m D C 4 C m=2 t 1 t 1 1 1 t.t t 2/ t t 2 C C Â 1 1 1 Ã t 1=2 t t m C C : D 4 t t 1 C Hence, by deﬁnition, the 2–loop polynomial is presented, modulo the equivalence (5), by

‚ KT .t ; t / Km 1 2 1 1 1 1 1 1 1 1 12m t1.t1 t 2/ t .t1 t 2/.t2 t /.t1t2 t t / : C 1 2 C 2 4 C 1 C 2 C 1 2 By symmetrizing this polynomial, we obtain the following proposition.

KT Proposition 2.4 The 2–loop polynomial of the Kinoshita–Terasaka knot Km is presented by

‚ KT .t1; t2/ m .2T1;0 2T2;0 2T2;1 T3;1/; Km D C

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where Tn;m ’s are deﬁned in (1), and its reduced 2–loop polynomial is presented by

2 2 ‚ KT .t/ 2m .t t /: y Km D C

We verify the proposition in Example 4.21 by using a surgery formula for the 2–loop polynomial. We can also verify that the special values ‚ KT .1/ ‚ KT . 1/ 4m y Km D y Km D satisfy Proposition 1.1, noting that

2m 1 1 1 1 V KT .t/ 1 .t 1/.t t 1/.t t /.t t 1/.t t 2/: Km D C C C C C C

2.4 The 2–loop polynomial of the Conway knot

In this section, we calculate the 2–loop polynomial for the Conway knot, from a surgery presentation of the knot. Similarly as the case of the Kinoshita–Terasaka knot in the previous section, we need, not the rational version, but the original version of the Aarhus integral to calculate the surgery in this case.

C The Conway knot Km (see[24]) is the knot given by

KC m ; m D

which is obtained from the Kinoshita–Terasaka knot by mutation. Similarly to the case of the Kinoshita–Terasaka knot, the Conway knot is obtained from the following link K L0 by 1=m surgery along the middle component L0 , [

K L0 ; [ D D

where we mean by a thick line 2 parallel copies of the line.

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We calculate the Kontsevich invariant of K L0 comparing it to the Kontsevich [ invariant of the link

K L [ 0 D

which is isotopic to the trivial 2–component link. In a similar way to the case of the 1 Kinoshita–Terasaka knot, we calculate Z.K L0/, where, instead of (18), we use [ an equivalent form of it, „

x y t −1 x y ! t /24 t x y x 1− t Â Ã 1 1 1 1 Z 1 : Á t C 96 C 96 48 t „ t /24 x y y

It follows that

− t t t t −1 t /24 − t /24 1 t /24 Z.K L0/ − t /24 − „ [ Á t/24 − 1−t − t − t/24

t 1 t 1 Â 1 1 1 Ã 1 : C C 32 C 24 48 t 1 t.t 1/ t 1

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Hence

1 1 Z.K L0/ Z.K L0/ „ [ „ [

− t t t t −1 t t −1 t /24 t /24 − t /24 −t/24 t /24 t /24 − t /24 − −t/24 − Á t/24 − t /24 − 1−t 1−t − − t − t − t/24 t /24

By Lemma 5.3, the ﬁrst diagram is equivalent, modulo diagrams of the form , to

−t −1 t −1 t t −1 t /24 t 1 −t/24 ! t /24 1 tt −t/24 − 1 t /24 − C 2 1−t t 1 − t − t /24

t −1

t t −1 −t −1 t /24 −t/24 t /24 −t/24 − .Á/ t /24 − 1−t − t − t /24

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1389 where the equivalence is obtained by Lemma 5.10. Hence

1 1 Z.K L0/ Z.K L0/ „ [ „ [ .Á/

− − −

− −1 1 t − t t t − −1 t 1 t /24 − t − t /24 − − t /24 t /24 − t /24 − − − − t /24 − − − t /24 t /24 t /24 − t /24 − − 1− t − 1− t − − − − t − t − − t /24 − t /24

Since the second diagram is related to the trivial link, the diagrams of the form in the second diagram cancel with each other. Hence, the degree 2 part of the ﬁrst diagram is calculated, as follows,

t −1 − t.t 1/ − t 1 − − − − t() t 1 t() t 1 t() t 1 t 1 − t t − 1 : −t −1 D t−1 −1 t−1 −1 D t−1 −1 C t

Therefore, t.t 1/ ! t() t − 1 1 1 t 1 Z.K L0/ Z.K L0/ t −1 1 : „ [ „ [ .Á/ t−1 −1 C

Since K L is isotopic to the trivial link, we have that [ 0 t.t 1/ ! 1 1 t 1 Z.K L0/ 1 : „ [ .Á/ C 48 C

Let L be L0 with 1=m framing. Then, similarly as in Section 2.1,

−1/2m ()− 2 1 t t 1 1=m 1 Z.K L/ − 1 L t 1 C [ Á − C 48 „ . / t 1 −1 t.t 1/ ! .m 1/.m 2/ t 1 : 48m C

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Further, from the deﬁnition of ,

1=2m t.t 1/ t.t 1/ t.t 1/ t 1 1 1Z.K L/ 1 t 1 t 1 t 1 L [ .Á/ t 1 C C 2 ! 2=m2 1 .m 1/.m 2/ C : C 48 48m

C The 2–loop part of the Kontsevich invariant of Km is given by the Aarhus integral, as follows, t.t 1/ 1Â t.t 1/ t 1 Ã2 1 .2–loop/ C t 1 t 1 Z .Km/ ; C D m=2 2 t 1 C 2 t2.t t 1 2/ t.t t 1 2/ Â C C 3 1 t 2.t t 1 2/ t 1.t t 1 2/ m C C D 2 t t 1 2 C t t 1 2 C C t3.t t 1 2/ t t 1 2 t.t 1/ C C Ã t t 1 2 1 t t 1 2 m t 1 C C C : t t 1 2 2 t t 1 2 C 2 t 1 1 C C Hence, by deﬁnition, the 2–loop polynomial is presented, modulo the equivalence (5), by

‚ C .t ; t / Km 1 2 3 1 1 1 1 1 2 2 1 3 1 12m .t1 t 2/.t2 t 2/.t1t2 t t 2/ t t t1t t C 1 C 2 C 1 2 2 1 2 C 2 1 2 6m t1.t1 1/.t2 1/.t1t2 1/: C C By symmetrizing this polynomial, we obtain the following proposition.

C Proposition 2.5 The 2–loop polynomial of the Conway knot Km is presented by

‚ C .t1; t2/ m .2T1;0 2T2;0 2T2;1 T3;1/ Km D C 3 1 1 2 2 1 2 1 2 m .t1t t t2 2/.t t2 t t 2/.t1t t t 2/ C 2 C 1 1 C 1 2 2 C 1 2 1 1 1 1 .t1 t 2/.t2 t 2/.t1t2 t t 2/ C 1 C 2 C 1 2 m .2T1;0 2T2;0 2T2;1 T3;1/ D C 3 1 2m T1;0 T2;1 T3;1 T4;0 T4;2 T5;0 T5;1 T6;2 T6;3 ; C C C C C 2 C

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1391

where Tn;m ’s are deﬁned in (1), and its reduced 2–loop polynomial is presented by

2 2 ‚ C .t/ 2m .t t /: y Km D C

We can verify that the special values ‚ C .1/ ‚ C . 1/ 4m satisfy Proposition 1.1, y Km D y Km D since the Conway knot and the Kinoshita–Terasaka knot are mutant and have the same Jones polynomial. Proposition 2.4 and Proposition 2.5 show that the 2–loop polynomial is sensitive to mutation, unlike the Alexander, Jones, HOMFLY and Kauffman polynomials.

3 The 2–loop polynomial for knots of genus 1

In this section, we give explicit presentations of the 2–loop polynomial for knots of genus 1 in Theorem 3.1 and Theorem 3.7. We give the presentations in Section 3.1 and prove Theorem 3.1 in Section 3.2–Section 3.4.

3.1 Presentation of the 2–loop polynomial for knots of genus 1

In this section, we give explicit presentations of the 2–loop polynomial for knots of genus 1, in terms of ﬁnite type invariants of a tangle which gives a part of a spine of a Seifert surface of the knot in Theorem 3.1, and in terms of derivatives of the Jones polynomial in Theorem 3.7.

Let T be a 2–component framed tangle, and let KT be the knot obtained from T as follows,

(2) (19) T T ; KT T ; D D

y where dotted lines in the picture of T imply strands possibly knotted and linked in some fashion, and T .2/ denotes the tangle obtained from T by replacing each component of T with 2 parallel copies of it. Any knot of genus 1 can be presented by KT for some T .

Geometry & Topology, Volume 11 (2007) 1392 Tomotada Ohtsuki

In particular, when T is of the following form (recalling that a boxed “n” implies the n full twists), its Kontsevich invariant is presented by

0 x 1 x n+k B C n/2 Â x y x x Ã B C 1 1 k (20) Z B −k C k 1 ; B C Á C 96 C 96 24 @ m+k A m/2 x y y y y y where we obtain the formula in a similar way as in Section 2.1.

Theorem 3.1 Let T and KT be as in (19). We present Z.T / by

Z.T / Á

x y x Â Â xx Ã Â yy Ã xy Â Ãx x Ã v2 1 v2 1 v2 k 1 v3 C 2 C 96 C 2 C 96 C 2 C 24 C x y y y y xx yy xy with some integers n; m; k; v2 ; v2 ; v2 ; v3 . Then, the 2–loop polynomial of KT is presented by nm 1 (21) ‚K .t1; t2/ .n m/.d / k.k /.k 1/ 12v3 T D C 2 C 2 C C 1 2 1 1 d.d 1/T1;0 d.d 1/T2;0 d d T2;1 2 C C C 3 3 xx yy 1 xy 12 mv nv .k /v 3v3 C 2 2 C C 2 2 C 2 1 1 2 2 1 d d T1;0 d T2;0 d d T2;1 ; C 6 C 2 3 2 where we put d nm k k , and Tn;m ’s are deﬁned in (1). D We give a proof of the theorem in Section 3.2.

Remark 3.2 Rozansky[40] conjectured that degree ‚K .t1; t2/ 2g.K/ for the t1 Ä genus g.K/ of K, which we prove in Theorem 4.7. In particular, we can verify this formula for knots of genus 1 concretely by Theorem 3.1; see also Table 1.

Remark 3.3 Rozansky[40] also conjectured that the 2–loop polynomial is a polynomial with integer coefficients. We show this for the knots of genus 1 by Theorem 3.1, as follows. Since the latter half of (21) has integer coefficients, it is sufficient to show that the former half has integer coefficients. Note that the first line of (21) is an integer. If d

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1393 is congruent to 1 modulo 3, then the second line of (21) has integer coefficients, and so 1 does ‚.KT /. Suppose that d is not congruent to 1 modulo 3. Since k.k /.k 1/ C 2 C is divisible by 3, it is sufficient to show that .n m/.d nm / is divisible by 3. Suppose C 2 that n m is not divisible by 3. Then, nm is congruent to 0 or 1 modulo 3. If nm is C congruent to 0, then d is congruent to 0. If nm is congruent to 1, then d is congruent nm to 2. In any case, d is divisible by 3. Therefore, ‚.KT / has integer coefficients. 2

An alternative presentation of the formula of ‚.KT / in Theorem 3.1 is that the 2–loop 1 part of log Z.KT / is given by t

.2–loop/ n m nm 1 1 Á Z .KT / C .d / k.k /.k 1/ v3 D 12 2 12 C 2 C C .t t 1 2/=.t/ .t t 1/=.t/ .t t 1 2/=.t/ Â C C Ã 3 3 1 Á d 4 2 C 4 .t t 1/=.t/ C .t t 1 2/=.t/ C .t t 1 2/=.t/ C 1 xx yy 1Á xy mv nv k v 3v3 ; 2 2 C 2 C 2 2 where we put .t/ 1 d.t t 1 2/, which is equal to the Alexander polynomial D C C of KT .

Recall that the Conway polynomial L.z/ ޚŒz (which we also denote by .L/) of r 2 r an unframed oriented link L is deﬁned by the skein relation Â Ã Â Ã Â Ã z r r D r

1=2 1=2 and the normalization .the trivial knot/ 1. Note that L.t t / K .t/. r D r D The scalars in the formula of Z.T / in Theorem 3.1 are elementarily calculated by the following proposition.

Proposition 3.4 Under the assumption of Theorem 3.1,

n .framing of T1/; m .framing of T2/; k lk.T1; T2/; D y D y D y y

xx 2 4 yy 2 4 .T1/ 1 v z O.z /; .T2/ 1 v z O.z /; r y D 2 C r y D 2 C 3 5 xx yy xy 2 4 .T1 T2/ kz 2v3z O.z /; .T / 1 .v v v /z O.z /; r y [ y D C r y D 2 C 2 C 2 C where T1 T2 and T are framed links given by y [ y y

Geometry & Topology, Volume 11 (2007) 1394 Tomotada Ohtsuki

T y1

T1 T2 T ; T T : y [ y D y D T y2

When we calculate the Conway polynomial in the proposition, we forget the framing. The proof of the proposition is an elementary exercise; for a reference, see eg[33]. Alternatively, in the linear skein modulo the relation

z ; D some formulas of Proposition 3.4 are rewritten

xx 2 4 T1 1 v z O.z / ; D 2 C yy 2 4 T2 1 v z O.z / ; D 2 C xx yy xy 2 4 3 5 T 1 .v v v /z O.z / kz 2v3z O.z / : D 2 C 2 C 2 C C C

Corollary 3.5 Under the assumption of Theorem 3.1, the reduced 2–loop polynomial is presented by ‚ .t/ y KT nm 1 2d 1 1 .n m/.d / k.k /.k 1/ 12v3 2 .t t 2/ D C 2 C 2 C C 3C C xx yy 1 xy 4 mv nv .k /v 3v3 K .t/; 2 C 2 C 2 2 T 1 where K .t/ 1 d.t t 2/ is the Alexander polynomial of KT . T D C C Proof By deﬁnition, we obtain ‚ .t/ from ‚ .t ; t / by replacing y K K 1 2 1 1 T1;0 2; T2;0 2.t t 2/; T2;1 t t 4: 7! 7! C C 7! C C Hence, we obtain the corollary from Theorem 3.1.

Example 3.6 For the pretzel knot K of type .p; q; r/, the reduced 2–loop polynomial and the Jones polynomial are given by

1 2d 1 1 Á ‚K .t/ .p q r/.4d 1/ pqr 2 C .t t 2/ ; y D 16 C C C C 3 C

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2 t t 1 p q r 1 p q r 1 p q p r q r VK .t/ 1 C C .t t t t t 1/; D .t 1/2 C C C C C C C C C C C where d .pq qr rp 1/=4; see Section 3.3 for a presentation of the 2–loop D C C C polynomial of the pretzel knot in terms of p, q, r . By the following formulas

1 ‚K .1/ .p q r/.4d 1/ pqr ; y D 8 C C C C 3 h 3 the coefﬁcient of h in VK .e / .p q r/.4d 1/ pqr ; D 8 C C C C 1 ‚K . 1/ .p q r/.4d 1/ pqr .1 4d/; y D 24 C C C C VK . 1/ K . 1/ 1 4d; D D V . 1/ 1 .p q r/.4d 1/ pqr; K0 D 2 C C C C we can verify the formula in Proposition 1.1.

Theorem 3.7 The 2–loop polynomial ‚K .t1; t2/ of a knot K of genus 1 is presented by

1 2 1 1 Á ‚K .t1; t2/ V .1/ 3V .1/ d d .T2;1 T1;0/ d.d 1/T2;0 D 24 K000 C K00 C 3 2 1 2 1 2 7 1 Á V . 1/ .5d 5d 1/T1;0 d.5d 1/T2;0 5d d T2;1 ; 16 K0 C C 2 3 C 3 1 1 where VK .t/ is the Jones polynomial of K, and d .1/ V .1/, and Tn;m ’s D 2 K00 D 6 K00 are deﬁned in (1).

Proof We choose a tangle T in (19) such that KT is isotopic to K. By the formula of Corollary 3.5 at t 1, we have that D ˙ nm 1 3 1 Á .n m/.d / k.k /.k 1/ 12v3 ‚K .1/ ‚K . 1/ C 2 C 2 C C D 4 y C 4d 1 y 1 V .1/ 3V .1/ 3 V . 1/; D 24 K000 C K00 C 2 K0 xx yy 1 xy 1 3 Á 4 mv nv .k /v 3v3 ‚K .1/ ‚K . 1/ 2 C 2 C 2 2 D 2 y C 4d 1 y 1 V .1/ 3V .1/ 9 V . 1/; D 36 K000 C K00 C 2 K0 where we obtain the second and fourth equality by Proposition 1.1, noting that VK . 1/ K . 1/ 1 4d . By substituting those formulas into the formula D D of Theorem 3.1, we obtain the required formula.

From the theorem, we obtain the following corollary.

Geometry & Topology, Volume 11 (2007) 1396 Tomotada Ohtsuki

Corollary 3.8 The reduced 2–loop polynomial ‚ .t/ of a knot K of genus 1 is y K presented by

1 1 1 1 ‚K .t/ .t t 2/ V .1/ 3V .1/ .t t 2/V . 1/VK . 1/; y D 72 C C K000 C K00 C 48 C K0 where VK .t/ is the Jones polynomial of K.

3.2 Proof of Theorem 3.1

The aim of this section is to prove Theorem 3.1. We reduce the proof to Proposition 3.9 below.

We denote by T0 the tangle in (20), that is, we put

x n+k n+k−k m+k − (22) T0 k ; KT ; D 0 D m+k

y for integers n; m; k .

Proposition 3.9 The 2–loop polynomial of the knot KT0 is presented by nm 1 ‚.KT / .n m/.d / k.k /.k 1/ 0 D C 2 C 2 C 1 2 1 1 d.d 1/T1;0 d.d 1/T2;0 d d T2;1 ; 2 C C C 3 3 2 where we put d nm k k as before, and Tn;m ’s are deﬁned in (1). D

We give two proofs of the proposition; one is a proof using the symmetry of KT0 , which we give in Section 3.3, and the other is a proof using a surgery presentation of

KT0 , which we give in Section 3.4.

Example 3.10 When k 0, the knot KT is isotopic to the . 4nm 1; 2n/ two- D 0 C bridge knot. Hence, by Proposition 2.3,

ˇ 1 ‚.KT0 /ˇk 0 nm.n m/ D D 2 C 1 2 2 1 1 nm.nm 1/T1;0 nm.nm 1/T2;0 n m nm T2;1 : 2 C C C 3 3 This is a particular value of the proposition.

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1397

Proof of Theorem 3.1 By deﬁnition, ‚.KT / is determined by Z.KT /. Further, from properties of the Kontsevich invariant, Z.KT / is determined by Z.T /. Hence, ‚.KT / can be determined by Z.T /. In particular, ‚.KT / does not depend on the degree > 2 part of Z.T /, because such a part can be presented by Jacobi diagrams with at least 3 trivalent vertices, and we can show, in the same way as the proof of Lemma 3.11 below, that such a Jacobi diagram changes Z.KT / by .> 2/–loop diagrams. Therefore, ‚.KT / can be determined by the degree 2 part of Z.T /. Ä

Let T0 be as in (22). Then,

x y x xx yy xy x x v2 v2 v2 Z.T / Z.T0/ v3 : Á C 2 C 2 C 2 C x y y y y Therefore, T0 is related to some T , satisfying that Z.T / Z.T / hence, ‚.KT / 0 Á 0 D ‚.K /, by clasper surgery along graph claspers of the form, T 0

x x x x

; ; ; :

y y y y

See Section 4.3 for graph claspers. Hence, by Lemma 3.11 and Lemma 3.12,

xx yy 1 xy 1 ‚.KT / ‚.KT / 6 mv nv .k /v .t1 t 2/ .t2/ .t3/ 0 2 C 2 C 2 2 C 1 3 1 1 1 1 1 v3 12 .t1 t /.t2 t / d .t1 t 2/.t2 t 2/ .t3/: C 4 1 2 C 4 C 1 C 2 By symmetrizing it, we have that

xx yy 1 xy ‚.KT / ‚.KT / mv nv .k /v D 0 2 C 2 C 2 2 2 1 1 2 2 1 12 d d T1;0 d T2;0 d d T2;1 C 6 C 2 3 2 1 2 1 2 4 1 v3 12 2d 2d T1;0 d d T2;0 2d d T2;1 : C C 2 C 2 C C 3 3 Therefore, by Proposition 3.9, we obtain the required formula.

A general form of the following lemma is given in Proposition 4.17.

Geometry & Topology, Volume 11 (2007) 1398 Tomotada Ohtsuki

Lemma 3.11 The changes of the 2–loop polynomial for surgery along the following graph claspers are presented, modulo the equivalence (5), by

0 1 0 1 B C B C − B C B n+kk m+k C 1 ‚ B − C ‚ B C 12 m .t1 t 2/ .t2/ .t3/; B n+k k m+k C @ A C 1 B C @ A

0 1 0 1 B C B C B C B C 1 ‚ B − C ‚ B C 12 n .t1 t 2/ .t2/ .t3/; B n+kk m+k C @ A C 1 B C @ A

0 1 0 1 B C B C B C B C 1 1 ‚ B − C ‚ B C 12 k .t1 t 2/ .t2/ .t3/: B n+k k m+k C @ A C 2 C 1 B C @ A

Proof We show the ﬁrst formula of the lemma. The ﬁrst knot of the formula is rewritten as

y n+k −k m+k x n+k −k m+k ; D D z w

where components depicted in thin lines imply surgery along the components. Hence, by Lemma 4.16, the difference of the Kontsevich invariant of the two knots in the ﬁrst formula is equivalent (modulo “ ”) to Á

n/2 k m/2 t−1 : x xx x xy y y x z y wz w

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1399

The equivariant linking matrix and its negative inverse are given by

0 n k 1 0 1 B k m 0 1 C M B C ; D @ 1 0 0 t 1A 0 1 t 1 1 0 0 1 1 1 m.t t 2/ t 1 k.t t 2/ 1 C 1 C C1 1 1 Bt 1 k.t t 2/ n.t t 2/ C M B C C C C ; D .t/ @ A where .t/ 1 .nm k2 k/.t t 1 2/, noting that we do not need the omitted D C C entries in this proof. Hence, the 2–loop part of the Kontsevich invariant of the result of the surgery is given by the rational version of the Aarhus integral as follows,

.t t 1 2/=.t/ x C ˛ ; m ; D x

x x y x y y where ˛ : D m.t t 1 2/=2 t 1 k.t t 1 2/ n.t t 1 2/=2 C C C C

The corresponding 2–loop polynomial gives the right-hand side of the ﬁrst formula of the lemma.

The other formulas of the lemma are obtained, in the same way, from

.t t 1 2/=.t/ .t t 1 2/=.t/ y C x C ˛ ; n ; ˛ ; k 1 : D D C 2 y y

A general form of the following lemma is given in Proposition 4.18.

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Lemma 3.12 The change of the 2–loop polynomial for surgery along the following graph clasper is presented, modulo the equivalence (5), by

0 1 0 1 B C B C B C B n+k−k m+k C ‚ B n+k−k m+k C ‚ B C B C @ A B C @ A

3 1 1 1 1 1 12 .t1 t /.t2 t / d .t1 t 2/.t2 t 2/ .t3/; 4 1 2 C 4 C 1 C 2 where d nm k2 k . D

Proof Similarly to the proof of Lemma 3.11, the left-hand side of the formula of the lemma is given by

m.t t 1 2/=.t/ .t 1 k.t t 1 2//=.t/ .t t 1/=.t/ x x C C C 1 ˛ ; ; 1 1 1 1 y y D n.t t 2/=.t/ .t 1 k.t t 2//=.t/ C2 .t t /=.t/ C C C 1 1 .t t /=.t/ .t t 2/=.t/ 3 1Á C d ; D4 .t t 1/=.t/ C 4 .t t 1 2/=.t/ C where ˛ is the one given in the proof of Lemma 3.11. The corresponding 2–loop polynomial gives the right-hand side of the formula of the lemma.

3.3 Proof of Proposition 3.9 by using symmetry

In this section, we give a proof of Proposition 3.9 using the symmetry of a pretzel knot, which is isotopic to KT0 .

The pretzel knot of type .p; q; r/ for odd integers p; q; r is given by

p/2 q/2 r/2 ;

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1401

where we mean k half twists by a boxed “k=2”. This is isotopic to KT0 given in (22) putting p 2n 2k 1, q 2k 1, r 2m 2k 1. We put D C C D D C C 0 1

P.p; q; r/ ‚ B p/2 q/2 C : D @ r/2 A

Then, the formula of Proposition 3.9 is rewritten as

(23) P.p; q; r/ 1 .p q r/.4d 1/ pqr D 16 C C C C 1 2 1 1 d.d 1/T1;0 d.d 1/T2;0 d d T2;1 ; 2 C C C 3 3 where d .pq qr rp 1/=4. The aim of this section is to prove this formula. D C C C Proof of Proposition 3.9 We show (23), which is equivalent to Proposition 3.9. Let P1.p; q; r/ be the coefﬁcient of T1;0 in P.p; q; r/. Putting

1 P1.p; q; r/ P1.p; q; r/ .p q r/.4d 1/ pqr d.d 1/ ; y D 16 C C C C we show that P1.p; q; r/ 0; the other part of (23) can be shown in the same way. y D By the symmetry of the pretzel knot, P.p; q; r/ (hence, P .p; q; r/) is a symmetric y1 polynomial in p; q; r . Hence, P .p; q; r/ can be presented by some polynomial y1 F.1; 2; 3/ in the elementary symmetric polynomials 1 p q r , 2 D C C D pq qr rp, 3 pqr . By comparing the pretzel knot with its mirror image, C C D P.p; q; r/ P. p; q; r/, hence, P1.p; q; r/ P1. p; q; r/. Therefore, D y D y F.1; 2; 3/ F. 1; 2; 3/. Further, by Lemma 3.15, D 2 r.r 1/ P1.p; q; r/ P1.p 2r; r; q 2r/ d.d 1/ ; D C C C 4 and hence, P1.p; q; r/ P1.p 2r; r; q 2r/: y D y C C Let ; ; be the elementary symmetric polynomials in p p 2r , q r , 10 20 30 0 D C 0 D r q 2r . Then, putting ˛ p q 2r , ˇ .p r/.q r/, we have that 0 D C D C C D C C 1 ˛ r; ˛ r; D 10 D C 2 2 ˇ r ; D 20 D 2 2 3 .ˇ ˛r r /r; .ˇ ˛r r /r: D C 30 D C C

Geometry & Topology, Volume 11 (2007) 1402 Tomotada Ohtsuki

It follows that F˛ r; ˇ r 2; .ˇ ˛r r 2/r F˛ r; ˇ r 2; .ˇ ˛r r 2/r: C D C C C Therefore, by Lemma 3.13 below, F. ; ; / (hence, P .p; q; r/) is equal to 0, as 1 2 3 y1 required.

Lemma 3.13 Let F.1; 2; 3/ be a polynomial in indeterminates 1; 2; 3 with rational coefﬁcients, satisfying that

F.1; 2; 3/ F. 1; 2; 3/; D F˛ r; ˇ r 2; .ˇ ˛r r 2/r F˛ r; ˇ r 2; .ˇ ˛r r 2/r; C D C C C where we also regard ˛; ˇ; r as indeterminates in this lemma. Then, F.1; 2; 3/ 0. D PN n 2 Proof Put F.1; 2; 3/ n 0 fn.1; 3/2 . Putting 2 0 and ˇ r , we have D D 2 D D 2 that f0.1; 3/ f0. 1; 3/ and f0 ˛ r;.2r ˛/r f0 ˛ r; .2r ˛/r . D D C C Hence, by Lemma 3.14 below, f0.1; 3/ 0. Since F.1; 2; 3/=2 satisﬁes the D assumption of the lemma, the lemma is shown by induction on N .

Lemma 3.14 Let G.1; 3/ be a polynomial in 1; 3 with rational coefﬁcients, satisfying that

G.1; 3/ G. 1; 3/; D G˛ r;.2r ˛/r 2 G˛ r; .2r ˛/r 2; D C C where we regard ˛; r as indeterminates in this lemma. Then, G.1; 3/ 0. D Proof From the assumption of the lemma, G˛ r;.2r ˛/r 2 is equal to a linear sum of r even˛odd with rational coefficients. Putting ˛ cr , it is a polynomial in D .c 1/r , .c 2/r 3 , which is equal to a linear sum of r oddcodd with rational coefficients. In particular, the coefficient of r k for each odd k is equal to a linear sum of .c 1/k ;.c 1/k 3.c 2/; .c 1/k 6.c 2/2; ; which is equal to a linear sum of codd . Hence, it is equal to 0.

Lemma 3.15

P.p; q; r/ P.p 2r; r; q 2r/ D C C 2 r.r 1/ 1 2 1 2 1 1 d.d 1/T1;0 d d T2;0 d d T2;1 : C 4 C 2 2 C C 3 3

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Proof Recall a presentation (22) of the pretzel knot of type .p; q; r/. We deform it by isotopy as follows,

j i k j i k i+1 k j D D

i+1 j k−1 i+1 k j ; D D

where these pictures present knotted framed graphs (with blackboard framing) such that by a graph we mean the knot of the boundary of a ribbon graph given by the framed graph. As in (19), the ﬁrst and the last terms are given by the following tangles,

x x i i+1 k ; j : − j k 1

y y

The Kontsevich invariant of them are given by (20) and by 0 1 x (i+j+1)/2 x y x x Â 1 1 j B C Z @ A j 1 Á C 96 C 96 24 y y (j+k)/2 x y y x x x j.j 1/ j .j 1/.2j 1/ Ã C C C ; 4 C 12 y y y

where we obtain the formula in a similar way as in Section 2.1; Lemma 3.16 and Lemma 5.3 are useful when we calculate the formula. Hence, by Lemma 3.11 and Lemma 3.12 and by putting j .r 1/=2, D P.p; q; r/ P.p 2r; r; q 2r/ D C C 2 r 1 2 1 1 2 2 1 3. r/ d d T1;0 d T2;0 d d T2;1 4 C 6 2 C 3 2 r.r 1/ 2 1 2 1 2 4 1 2d 2d T1;0 d d T2;0 2d d T2;1 : C 4 C 2 C 2 C C 3 3 This gives the required formula.

Geometry & Topology, Volume 11 (2007) 1404 Tomotada Ohtsuki

Lemma 3.16 For scalars b and c,

c c bc bc2 x y Á 1 x y : b b x y x y Á x y 2 C 2

Proof By induction on the number of chords labeled by c, we can show that

c c c bc bc2 Á : b Á b C 2 C 2

The required formula can be shown from the above formula by induction on the number of chords labeled by b. A detailed proof is left to the reader.

3.4 Proof of Proposition 3.9 from surgery presentations

In this section, we give another proof of Proposition 3.9 by using a surgery presentation of KT0 . Indeed this proof might be somehow tedious comparing to the previous proof, but the way of this proof is generalized to the case of higher genus later in Section 4.

Proof of Proposition 3.9 The knot KT0 given in (22) is obtained from the following link by surgery along the components of thin lines. This link is isotopic to the second link, which we denote by K L, where K denotes the knot of thick line and L denotes [ the link of thin lines.

x y n+k −k m+k n+k −k m+k

z K L D w D [

We calculate the Kontsevich invariant of K L by decomposing it into the following [ parts; each part can be calculated in a similar way as in Section 2.1,

0 x 1 y n/2 m/2 Â x y x x Ã − 1 1 k Z @ n+k k m+k A k 1 ; D C96 C96 24 x y x y y y

x 0 x 1 x z x x Â 1 1 1 Ã Z @ z A 1 ; D C 96 C 96 24 x z z z z

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1405

z z w w 0 1 z z z w Â Ã − 1 Z 1 1 ; @ z w A D C 24 w w

0 1 z w z w Â z z Ã 1 z w t 1 1 1 t t Z @ A 1 : D C96 C96 24 „ z w w w

By composing them, we obtain Z.K L/. Since Z.K L/ is obtained from Z.K L/ [ L [ [ by connect-summing to each component of L by deﬁnition, we have that

n/2 m/2

x k 1 y Z.K L/ .1 ˇ1/; „ L [ D C −1 z t w where x y z w x x x x y y z z z z 1 1 1 1 k 1 1 1 1 t t 12ˇ1 : D 2 C 2 C 2 C 2 2 2 2 C 2 2 x y z w y y z z w w w w w w

Further, by Lemma 5.2, n/2 m/2 x x y x k Â 2 y k 1 1 1 Á C 8 C 8 C 8 y z w −1 z 3 x x x x y y t w k 1 1 Ã : 12 y y 12 z z 12 w w Hence, by Lemma 5.15,

n/2 m/2 x x k y y x y y 1 x ;x;yZ.K L/ .1 ˇ1 ˇ2/; L [ .Á~/ −1 C C „ z t w

x y x x y n2 m2 3k2 nk mk n 3 m 3 where 12ˇ2 C C C C D 4 C 4 C 2 C 2 C 2 x y y z w x x x x y y x y x y k k k3 : y y z z w wC 2 z z C 2 w w

Geometry & Topology, Volume 11 (2007) 1406 Tomotada Ohtsuki

Further, by Lemma 5.17, Lemma 5.19, and Lemma 5.20,

n=2 k m=2 t 1 1 Z.K L/ .1 ˇ1 ˇ2 ˇ3/; L [ .Á~/ x x x y y y x z y w z w C C C z t 1 z z z z z y 3 C t t where 12ˇ3 3 D2 C t w w w w w w w x x y y z z z z t 1 1 1 1 − 2t−1 2t 1 : t 1 t 1 t C 2 z w C 2 z w C 2 w C 2 w w w

The equivariant linking matrix M is given by

0 n k 1 0 1 B k m 0 1 C 1 1 M B C ; M FXY .t/ ; D @ 1 0 0 t 1A D .t/ 0 1 t 1 1 0

2 1 where .t/ 1 .nm k k/.t t 2/ and FXY .t/ is given by D C C

1 1 Fxx.t/ m.t t 2/; Fxy.t/ t 1 k.t t 2/; D C D C C 1 Fxz.t/ 1 k.t 1/; Fxw.t/ m.t 1/; D C D 1 1 Fyy.t/ n.t t 2/; Fyz.t/ n.t 1/; D C D Fyw.t/ 1 k.t 1/; Fzz.t/ n; D C D 2 Fzw.t/ k .nm k /.t 1/; Fww.t/ m; D C D

1 1 1 and FYX .t/ FXY .t /. Therefore, we obtain Z.KL/ from Z.K L/ by D L [ surgery along L using the rational version of the Aarhus integral,

Z 1 1 ˝ ˛ Z.KL/ Z.K L/ dx dy dz dw ˛; .1 ˇ1 ˇ2 ˇ3/ D L [ Á C C C where

Y YX X YX Á (24) ˛ exp D t D t X;Y x;y;z;w FXY .t/=2 X;Y x;y;z;w FXY .t/=2 D D

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1407

x y z w n2 2 m2 2 1 1 12.ˇ1 ˇ2 ˇ3/ C C C C D 4 C 4 C 2 C 2 x y z w x x y z 3k2 nk mk n 3 m 3 3 t 1 C C C C C C 2 C 2 C 2 C 2 y z w w x x x x y y z z z z z y 3 k 3 3 3 3 t t .k / 3 t C 2 y y 2 z z 2 w wC 2 w w 2 w w w w x x y y x y x y z z z z 1 1 k k 1 t 1 − t 1 t 1 2t−1 2t 1 : C 2 C 2 C 2 C 2 C 2 C 2 t z w z w z z w w w w w w

We calculate the corresponding part of each diagram of 12.ˇ1 ˇ2 ˇ3/ in the 1 C C formula of Z.KL/ as follows. Noting that, by deﬁnition, F .t/=.t/ X XY ˛; D Y for X; Y x; y; z; w, we have that D x y ˛; 3m ; ˛; 2n ; D D x y x z ˛; .1 2k/ ; ˛; n ; D C D y z w x y ˛; m ; ˛; ˛; k ; D D D C w z w z t 1 Á ˛; C k 2 ; D C w where, in this section, we deﬁne the markings by

1 1 t t 2 t t 1 C2.t/ 2.t/ .t/ (25) ; ; : D D D

Further, noting that, by deﬁnition,

FXY .t/=.t/ FXW .t/=.t/ .FXZ .t/ FZX .t//=.t/ XY 1 ˛; D C2 ZW F .t/=.t/ F .t/=.t/ .FYW .t/ FWY .t//=.t/ ZW ZY

Geometry & Topology, Volume 11 (2007) 1408 Tomotada Ohtsuki for X; Y x; y; z; w, we have that D x x ˛; .4d 1/ 3 ; y y D x x y y ˛; ˛; 2.d k2/ k2 3k2 ; z z D w w D C z z Á ˛; .d k/ 2k .d k/ 3.d k/ ; w w D C C C x y Á ˛; k n 2 3 ; z z D C C x y Á ˛; km 2 3 ; w w D C C z z t t ˛; .d k/ 2kd d 2 3d 2 ; w w D C C z y Á ˛; t k d 3d ; w w D C C x x Á ˛; t 1 2m 2d ; z w D C y y Á ˛; t 1 2n 2d ; z w D C z z z t Á ˛; 2t−1 n. d k/ 2 3 ; D C C C w z Á ˛; 2t−1 m. d k/ 2 3 : t D C C C w w w

Hence, the 2–loop part of the Kontsevich invariant of KL is presented by

.2–loop/ ˝ ˛ 12 Z .KL/ ˛; 12.ˇ1 ˇ2 ˇ3/ 1 2 D 1 2 C C 1 1 2 m.n 2/ 2 n.m 2/ 2 n 2 m D C C C C Á 1 .3k2 nk mk/.1 2k/ 1 .n 3 m 3/ k C 2 C C C C 2 C C C C Á Á 3 k 2 .k3 1 k/ .4d 1/ 3 C 2 C C 2 Á 3 2.d k2/ k2 3k2 C C C C C Á 3 .d k/ 2k.d k/ .d k/2 3.d k/2 C 2 C C C C

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1409

Á 3 .d k/ 2kd d 2 3d 2 2 C C Á Á 3k d 3d 1 .n m/ 4d 2 C C C 2 C C Á Á 1 .n m/k2 2 3 1 .n m/.k d/ 2 3 C 2 C C C C 2 C C C Á ..n m/.d 1 .nm// k.k 1 /.k 1// 3 .4d 1/ 3 D C 2 C 2 C C .n m/.d 1 .nm// k.k 1 /.k 1/ D C 2 C 2 C 1 1 1 .t t 2/=.t/ .t t /=.t/ .t t 2/=.t/ Â 3 C 3 1Á C Ã d : 2 C 4 .t t 1/=.t/ C 4 .t t 1 2/=.t/ C

This gives the formula of Proposition 3.9.

4 The 2–loop polynomial for knots of any genus

In this section, we give a presentation of the 2–loop part of the Kontsevich invariant for knots of any genus. By using the presentation, we show that the degree of the 2–loop polynomial of a knot is bounded by twice the genus of the knot in Section 4.2. Further, we show clasper surgery formulas for the 2–loop polynomial in Section 4.3.

4.1 The 2–loop polynomial for knots of genus 2

Before we calculate the case of any genus, we calculate the case of genus 2 in this section. The approach of the calculations for both cases are almost the same. Similarly to the case of genus 1, we let T be a 4–component framed tangle, and let KT be the knot obtained from T as follows,

T(2)

T ; KT ; D D x1 y1 x2 y2 where dotted lines in the picture of T imply strands possibly knotted and linked in some fashion, and T .2/ denotes the tangle obtained from T by replacing each component of T with 2 parallel copies of it. Any knot of genus 2 can be presented by KT for some T .

Geometry & Topology, Volume 11 (2007) 1410 Tomotada Ohtsuki

We consider a particular form T0 of T for integers mi and kij ; recall that a boxed “k ” implies k full twists:

−k14

−k13 −k24

(26) T0 D m1m2 m3 m 4 −k12 −k23 −k34

Note that ‚.KT / for any T can be obtained from ‚.KT0 / by Proposition 4.3; the proposition does not change the linking matrix of T , while T0 represents a class of T having an arbitrarily ﬁxed linking matrix.

By a similar argument as in Section 3.4, we obtain KT from the following link K L 0 [ by surgery along L:

−k14

x2 y1 x1 y −k13 −k24 2

K L m1 m2 m3 m4 [ D −k12 −k23 −k34

z1 w2

w1 z2 where K denotes the knot of thick line and L denotes the link of thin lines.

We calculated the Kontsevich invariant of K L, in a similar way as in Section 3.4, [ as follows. We rename the labels by X1 x1 , X2 y1 , X3 x2 , X4 y2 , and D D D D Z1 z1 , Z2 w1 , Z3 z2 , Z4 w2 , and use both names in formulas of this D D D D section, to simplify their presentations. By a similar argument as in Section 3.4, the

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1411 upper part of 1Z.K L/ is given by L „ [

w1 z1

y1 x1 n2/2 n1/2 k12 Xi X X k24 Â Xi Xi 2 1Ã X 1 X kij k13k24 k23 k14 1 ; k13 C 24 24 C 2 1 i 4 X 1 i

z2w 2

P 1 where ni mi kij putting kij kji . Further, the middle part of Z.K L/ D j i D L [ is given by ¤ „

Xi X X Y Â X Â1 1 i iÃÃ 1 : t X Z C 8 8 1 i 4 i i 1 i 4 Z Zi Zi Ä Ä Ä Ä i Hence, by Lemma 5.28, 1Z.K L/ of the above parts is given by L [ n =2 kij Xi Xl Y i Y Y Y ij l .1 ˇ1/; t X X t X X t X Z t C 1 i 4 i i 1 i

Geometry & Topology, Volume 11 (2007) 1412 Tomotada Ohtsuki

The lower part of 1Z.K L/ is calculated similarly as in Section 3.4, but a different L [ point is to use „

0 1 z1 w1 z2 w2 w z2 − 1 − t 1 z t t t t 1 1 w2 2 Z @ A „ Á z w Â Â i i zi zi ÃÃ 1 X t t 1 ; C 24 C 1 i 2 zi wi wi wi Ä Ä which is obtained from ! − t −1 1 2 Z .1/ ; „ Á where we write 1 2 if 1 2 is equal to a linear sum of diagrams with at least Á.1/ 1 trivalent vertex. It follows that the lower part of 1Z.K L/ is given by L „ [ x yx y 1 1 2 2 z w Â Â i i zi zi zi ziÃ z1 z2 Ã − − 1 X t t 1 t t 1 1 1 t 1 : t t C 24 C C w w w w C 2 w w z wz w 1 i 2 zi wi i i i i 1 2 1 1 2 2 Ä Ä Hence, by Lemma 5.17, 1Z.K L/ of the lower part is given by L [ Â t 1 zi zi zi Ã Y t=2 t=2 .1 ˇ2/; z w C 1 i 2t i i wi wi wi Ä Ä where z w Â i i zi zi zi zi zi zi zi zi zi X 1 1 3 3 3 t t t − − t 12ˇ2 t 1 t 1 3 3 t D 2 C2 C2 C2 w w 2 w w w wC w w 1 i 2 zi wi wi i i i i i i i i Ä Ä zi zi wi zi xi xi yi yi Ã z1 z2 t 1 t 1 1 tt 1 t C 6 : t t 1 2 t 1 2 t 1 C zi wi C wi wCi C zi wi C zi wi C w1 w2 By composing the above resulting formulas for the parts of 1Z.K L/, we have L [ ni=2 kij Xi Xl 1 Y Y Y Y Z.K L/ ij l L [ .Á~/ t X X t X X t X Z t 1 i 4 i i 1 i

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1413

Its equivariant linking matrix is presented by ! A I M ; D I J where I is the unit matrix of size 4, and A is the linking matrix of T0 , and 0 0 t 1 0 0 1 1 Bt 1 0 0 0 C J B C: D 0 0 0 t 1 @ A 0 0 t 1 1 0 1 1 As in Section 3.4, we obtain Z.KT / from Z.K L/ by the rational version 0 L [ of the Aarhus integral; in particular, the 2–loop part is presented by .2–loop/ ˝ ˛ Z .KT / ˛; .ˇ1 ˇ2 ˇ3/ ; 0 D C C where Q Q Y P Â X P Ã ˛ exp ; D t D t P;Q Xi ;Zj FPQ.t/=2 P;Q Xi ;Zj FPQ.t/=2 D D Â Xi Xl Â Zi Zi Zi ÃÃ2 X 1 X t ˇ3 ; D ijl C 2 C t 1 i

In this section, we give a presentation of the 2–loop part of the Kontsevich invariant for knots of any genus g. By using the presentation, we show that the degree of the 2–loop polynomial of a knot is bounded by twice the genus of the knot. Similarly to the case of genus 2, we let T be a 2g–component framed tangle, and Ä let KT be the knot obtained from T , as follows,

T ; D XXXX1 2 3 4 XX2g −1 2g

T(2)

(27) KT ; D

Geometry & Topology, Volume 11 (2007) 1414 Tomotada Ohtsuki where dotted lines in the picture of T imply strands possibly knotted and linked in some fashion, and T .2/ denotes the tangle obtained from T by replacing each component of T with 2 parallel copies of it. Any knot of genus g can be presented by KT for some T .

Similarly to the case of genus 2, we let T0 be a particular tangle of T such that its linking matrix is .kij /, where we put kii ni , and any 4–component sub-tangle of T0 D is of the form (26). Further, we set K L as in Section 4.1, and label the components [ of L by xi; yi; zi; wi similarly to the case of genus 2, and rename them by X2i 1 xi , D X2i yi , Z2i 1 zi , and Z2i wi . D D D In the same way as in Section 4.1, we obtain the following proposition.

Proposition 4.1 The 2–loop part of the Kontsevich invariant of KT0 is given by

.2–loop/ ˝ ˛ Z .KT / ˛; .ˇ1 ˇ2 ˇ3/ ; 0 D C C where Q Y P ˛ for FPQ.t/ given in Lemma 4.2, D t P;Q Xi ;Zj FPQ.t/=2 D X X Â 2 i i Xi XiÃ X ni 2 ni 3 3 12ˇ1 C C D 4 C 2 2 1 i 2g X Z Zi Zi Ä Ä i i X Â 2 i Xi Xi Xi Xj Xi Xj Ã X 3kij nikij nj kij kij Á kij kij C C k3 C 2 ij C 2 C 2 C 2 1 i

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1415

z w Â i i zi zi zi zi zi zi zi zi zi X 1 1 3 3 3 t t t − − t 12ˇ2 t 1 t 1 3 3 t D 2 C 2 C 2 C 2 w w 2 w w w wC w w 1 i g zi wi wi i i i i i i i i Ä Ä zi zi wi zi xi xi yi yi Ã zi zj t 1 t 1 1 X tt 1 t t C t 1 t 1 t 1 6 ; C C CC 2 C 2 C zi wi wi wi zi wi zi wi 1 i

Lemma 4.2 The FPQ.t/’s in Proposition 4.1 are presented by

FXi Xj .t/ 1=2 1=2 T 1=2 1=2 T 1 .t t / e .t V t V / ej ; .t/ D i Â 1=2Ã g FXi Zj .t/ T 1=2 1=2 T 1 0 t ˚ e .t V t V / ej ; .t/ D i t 1=2 0 Â 1=2Ã g FZi Xj .t/ T 0 t ˚ 1=2 1=2 T 1 e .t V t V / ej ; .t/ D i t 1=2 0 Â 1=2Ã g FZi Zj .t/ T 0 t ˚ 1=2 1=2 T 1 e .t V t V / A ej ; .t/ D i t 1=2 0 where A is the linking matrix of T , and V is the Seifert matrix of a natural Seifert 1=2 1=2 T surface of KT in (27), and .t/ det t V t V . Further, for P; Q; R; S D 2 Xi; Zj and ˛ given in Proposition 4.1: f g F .t/=.t/ P PQ ˛; D Q

FPQ.t/=.t/ FPS .t/=.t/ .FPR.t/ FRP .t//=.t/ PQ 1 ˛; D C2 RS F .t/=.t/ F .t/=.t/ .F .t/ F .t//=.t/ RS RQ QS SQ

1 Proof In a similar way to Section 4.1, FPQ.t/’s are the entries of FPQ.t/ M , D where ! Â 1=2Ã g A I 0 t ˚ M ; J ; 1=2 1=2 1=2 D I .t t / J D t 0

Geometry & Topology, Volume 11 (2007) 1416 Tomotada Ohtsuki and I is the unit matrix of size 2g. Hence,

1=2 1=2 1 1 ! 1 1 .t t / S S J M ; D .t/ JS 1 JS 1A 1=2 1=2 T g where we put S t V t V and .t/ det S , noting that V A 0 1 ˚ . D D D C 0 0 Therefore, we obtain the former four formulas of the lemma. We obtain the latter two formulas from the deﬁnition of the bracket.

Proposition 4.3 Let T , T0 , and KT be as above, such that the linking matrices of T 1 and T0 are equal. We present Z.T / by 1 1 log Z.T / log Z.T0/ t Á t X X Xi X X X i l X X i j a bij c : C ijk C C ijlh 1 i

Xi X X X X i j ˇ bij c 10 D C ijlh 1 i

Proof As in Section 4.1, KT (resp. KT ) is obtained from a link K L (resp. 0 [ K0 L0 ) by surgery along L (resp. L0 ). By Lemma 5.24, the Kontsevich invariant [ of K L and K0 L0 are related by [ [ 1 1 log Z.K L/ log Z.K0 L0/ t L [ t L [ X X Xi X X X i l X X i j a bij c : Á ijk C C ijlh 1 i

1 1 Since Z.K L/ Z.K0 L0/ L [ Á.2/ L [

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1417

X X Z Z Z X i l 1 X i i i 1 t ; Á.2/ C ijl C 2 C t 1 i

Theorem 4.4 (see a problem in[19]) For a tangle T and the knot KT given in (27), the 2–loop polynomial ‚.KT / of KT can be presented by ﬁnite type invariants of T of degree 3. In other words, the 2–loop polynomial of a knot can be presented by Ä ﬁnite type invariants of degree 3 of a knotted trivalent graph which is a spine of a Ä Seifert surface of the knot.

Proof By Proposition 4.3, the 2–loop part of the Kontsevich invariant of KT is presented by ﬁnite type invariants of T of degree 3, hence, so is the 2–loop polynomial Ä of KT . This is the former statement of the theorem.

We put the knotted trivalent graph GT by

GT : D

Then, GT is related to a spine of a Seifert surface of the knot by a sequence of the local move . Further, when two knotted trivalent graphs are related by this $ move, ﬁnite type invariants of one of them can be presented by ﬁnite type invariants of the other. Hence, we obtain the latter statement of the theorem.

Remark 4.5 The 2–loop polynomial of the knot given as the boundary of the ribbon graph of a knotted trivalent graph is a ﬁnite type invariant of the knotted trivalent graph of degree 3 which is unchanged under the local move . In general, the Ä 1 $ n–loop part of log Z.K/ can be presented by ﬁnite type invariants of degree .2n 1/ of the knottedt trivalent graph which is a spine of a Seifert surface of the Ä knot such that the invariant is unchanged under the local move . $

Geometry & Topology, Volume 11 (2007) 1418 Tomotada Ohtsuki

Remark 4.6 Rozansky[40] conjectured that the coefficients of the 2–loop polynomial 1 are integers. Marche´[26] proved a weaker statement that the coefficients are in 12 ޚ. From Proposition 4.1, we can obtain an integrality result, but it is a further weaker 1 statement that the coefficients are in 24 ޚ.

Theorem 4.7 (a conjecture in[40]) The degree of the 2–loop polynomial of a knot K is bounded by twice the genus g of K, degree ‚K .t1; t2/ 2g: t1 Ä

Proof We choose a tangle T whose KT in (27) is isotopic to K. By Proposition 4.1 and Proposition 4.3, it is sufﬁcient to show that ˝ ˛ (28) ˛; .ˇ1 ˇ2 ˇ3 ˇ0 ˇ0 / 0; 1 3 .deg 2g/ C C C C ÁÄ where ˛; ˇ1; ˇ2; ˇ3; ˇ10 ; ˇ30 are given in the propositions, and we write

1 2 .deg 2g/ ÁÄ if the t1 –degree of the corresponding 2–loop polynomial of 1 2 is at most 2g. We calculate the ˇ1 ˇ part of (28) as follows. For each diagram of ˇ ˇ , by C 10 C 10 Lemma 4.2, its corresponding part in (28) is equal to a linear sum of diagrams of the form f .t/=.t/ g.t/=.t/

where f .t/ and g.t/ are either FXi Xj .t/, FXi Zj .t/, FZi Xj .t/, or FZi Zj .t/. The corresponding 2–loop polynomial of this form is

X " " " 1 1 1 f .t /g.t /.t / ޑŒt ; t ; t =.t1t2t3 1/: i j k 2 1˙ 2˙ 3˙ D " 1 i;j;kD˙1;2;3 f gDf g By Lemma 4.8, its t1 –degree is at most 2g. Hence, ˝ ˛ ˛; ˇ ˇ0 0: 1 .deg 2g/ C ÁÄ

We calculate the ˇ2 part of (28) as follows. By the same argument as above, diagrams 1 without labels of polynomials in t ˙ vanish in (28). For the other diagrams, the

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1419 corresponding parts in (28) are calculated as follows,

Fzi wi .t/=.t/ Fzi wi .t/=.t/ zi t 1 t ˛; t−1 t−1 2 ; .deg 2g/ D t 1 ÁÄ wi zi zi t t ˛; 0; .deg 2g/ wi wi ÁÄ Fzi wi .t/=.t/ zi zi ˛; t t ; .deg 2g/ wi wi ÁÄ Fwi zi .t/=.t/

Fzi wi .t/=.t/ zi zi ˛; t ; t .deg 2g/ 2 wi wi ÁÄ t Fwi zi .t/=.t/ zi zi wi zi t 1 t ˛; t C ˛; t 1 0; .deg 2g/ .deg 2g/ zi wi ÁÄ wi wCi ÁÄ xi xi yi yi ˛; t 1 ˛; t 1 0; .deg 2g/ .deg 2g/ zi wi ÁÄ zi wi ÁÄ 1 t Fzi wj .t/=.t/ zi zj tt 1 t t 1 ˛; 0: .deg 2g/ 1 .deg 2g/ wi wj ÁÄ t Fwi zj .t/=.t/ ÁÄ Hence ˝ ˛ ˛; ˇ2 .deg 2g/ ÁÄ Fzi wi .t/=.t/ Fzi wi .t/=.t/ Fzi wi .t/=.t/ 1 X Â Ã t t : 4 C F .t/=.t/ t2F .t/=.t/ 1 i g wi zi wi zi Ä Ä The ﬁrst and third diagrams are further calculated as follows,

Fzi wi .t/=.t/ Fzi wi .t/=.t/ t g 1 .g/ t C =.t/ ; .deg 2g/ ÁÄ

Fzi wi .t/=.t/ Fzi wi .t/=.t/ ; 2 .deg 2g/ g 1 .g 1/ t Fwi zi .t/=.t/ ÁÄ t C Fwizi =.t/

.g/ g .g 1/ where we denote by the coefﬁcient of t in .t/ and denote by Fwi zi the g 1 coefﬁcient of t in Fwi zi .t/. Since they cancel with each other by Lemma 4.9, we

Geometry & Topology, Volume 11 (2007) 1420 Tomotada Ohtsuki have that

Fzi wi .t/=.t/ ˝ ˛ 1 X t ˛; ˇ2 : .degÁ2g/ 4 F .t/=.t/ Ä 1 i g wi zi Ä Ä We calculate the ˇ3 ˇ part of (28) as follows. By the same argument as in the case of C 30 ˇ1 , most diagrams in ˇ3 ˇ vanish in (28). The surviving case is graphically shown C 30 as follows. zi wi

Fz w (t) z i i w t/2 i i t/2

wi zi Hence, we have that

Fzi wi .t/=.t/ Fzi wi .t/=.t/ 1 2 1 2 ˝ ˛ X t Fzi wi .t/=.t/ X t Fwi zi .t/=.t/ ˛; ˇ3 ˇ30 C .degÁ2g/ 4 .degÁ2g/ 4 Ä 1 i g Fwi zi .t/=.t/ Ä 1 i g Fwi zi .t/=.t/ Ä Ä Ä Ä Therefore, by summing the resulting formulas of the ˇ1 ˇ ; ˇ2; ˇ3 ˇ cases, we C 10 C 30 have that ˝ ˛ ˛; .ˇ1 ˇ10 ˇ2 ˇ3 ˇ30 / C C C C F .t/=.t/ F .t/=.t/ zi wi zi wi ! 1 2 X t t Fwi zi .t/=.t/ .degÁ2g/ 4 C Ä 1 i g Fwi zi .t/=.t/ Fwi zi .t/=.t/ Ä Ä F .t/=.t/ F .t/=.t/ zi wi zi wi ! 1 X tg 1.g/=.t/ tg 1F .g 1/=.t/ C C wizi : .degÁ2g/ 4 C Ä 1 i g Fwi zi .t/=.t/ Fwi zi .t/=.t/ Ä Ä Since this vanishes by Lemma 4.9, we obtain (28), as required.

Lemma 4.8 The minimal and maximal degrees of polynomials given in Lemma 4.2 are bounded, as follows, min-deg .t/ g; max-deg .t/ g; Ä min-deg FX X .t/ g; max-deg FX X .t/ g; i j i j Ä min-deg Fz z .t/ .g 1/; max-deg Fz z .t/ g 1; i j i j Ä min-deg Fw w .t/ .g 1/; max-deg Fw w .t/ g 1; i j i j Ä min-deg Fz w .t/ .g 1/; max-deg Fz w .t/ g; i j i j Ä min-deg Fz X .t/ .g 1/; max-deg Fz X .t/ g; i j i j Ä min-deg Fw X .t/ g; max-deg Fw X .t/ g 1: i j i j Ä

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1421

Further, if i j , then min-deg Fz w .g 2/. ¤ i j Proof The bound for .t/ is obtained from the fact that .t/ det.t 1=2V T t 1=2V / D and V is a .2g/ .2g/ matrix.

The bound for FXi Xj .t/ is obtained as follows. From its definition in Lemma 4.2, 1=2 1=2 T 1=2 1=2 T 1 FX X .t/ .t t / e .t/.t V t V / ej : i j D i The second factor in the right-hand side is equal to the .i; j / entry of .t/.t 1=2V t 1=2V T / 1 , which is equal to a minor determinant of .t 1=2V t 1=2V T /. Hence, we obtain the required bound for FXi Xj .t/. In the same way, we obtain the bounds of the other polynomials except for the last one. The last bound of the lemma is obtained as follows. By definition, 1=2 T 1=2 1=2 T 1 Fzi wj .t/ FZ2i 1Z2j .t/ .t/ t e2i .t V t V / A e2j D D 1=2 T 1=2 1=2 T t e .t V t V / A e2j ; D 2i 0 1 where we denote .det M /M by M 0 for a regular matrix M , noting that entries of M 0 are presented by minor determinants of M . Hence, similarly as in the above . g 1/ case, we have that min-deg Fzi wj .t/ .g 1/. Further, the coefficient FziwjC of g 1 t C in Fzi wj .t/ is calculated as follows, . g 1/ T T T T T T F e . V / A e2j e . V / V e2j .det V / e e2j 0: ziwjC D 2i 0 D 2i 0 D 2i D This implies the required bound.

.g 1/ .g/ .g/ g Lemma 4.9 We have that Fwi zi , where denotes the coefﬁcient of t .g 1/ D g 1 in .t/, and Fwi zi denotes the coefﬁcient of t in Fwi zi .t/.

Proof In the same way as in the proof of Lemma 4.8, .g 1/ T F .det V / e e2i .det V /: wi zi D 2i D Further, .g/ det V , since .t/ det.t 1=2V t 1=2V T /. Hence, we obtain the D D lemma.

Remark 4.10 The bound of Theorem 4.7 might be sharp for most knots as we see in Table 1, Table 2, Example 4.11 and Example 4.13. However, we can construct examples which give inequality of the formula of Theorem 4.7. Such examples can be constructed by surgery along graph claspers with at least 3 trivalent vertices; for graph

Geometry & Topology, Volume 11 (2007) 1422 Tomotada Ohtsuki claspers, see Section 4.3. More generally, it is known[7] that we can construct knots which have the same .< n/–loop part of the Kontsevich invariant, but have different n–loop part, for any n. For example, the following knot and the trivial knot give such examples, when this graph clasper has 2n trivalent vertices.

In particular, this knot gives the inequality of the formula of Theorem 4.7.

Example 4.11 It is shown in[35] that the degree of the 2–loop polynomial of the torus knot of type .p; q/ equals .p 1/.q 1/. Since the genus of the torus knot of type .p; q/ equals .p 1/.q 1/=2 (see eg[24]), torus knots give the equality of the formula of Theorem 4.7.

Example 4.12 In Theorem 3.7, we gave a presentation of the 2–loop polynomial for knot of genus 1. Its degree satisﬁes the formula of Theorem 4.7, and its equality holds when V . 1/ 0 or v3.K/ 0 (see also Proposition 1.1). K0 ¤ ¤ KT Example 4.13 The generalized Kinoshita–Terasaka knot Km;n [16] and the general- C ized Conway knot Km;n (see[24]) are given by

− n+1 n+1 − n+1 − 2 2 2 n/2

KKT m ; KC m ; m;n D m;n D − n+1 n/2 n/2 n/2 2 where a boxed “k=2” implies k half twists, as before. We show in Proposition 4.14 and Proposition 4.15, that the degree of the 2–loop polynomial for these knots are 2n 1 and 4n 2 respectively. This implies, by Theorem 4.7, that their genera are at least n and 2n 1. Since there exist their Seifert surfaces of these genera, it follows that these are exactly their genera, as it has been shown in[8] geometrically, and in [38] by using the knot Floer homology.

Proposition 4.14 The t –degree of the 2–loop polynomial ‚ KT .t ; t / of the gen- 1 Km;n 1 2 eralized Kinoshita–Terasaka knot KKT is equal to 2n 1. m;n

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1423

Proof For n 2, a concrete presentation of ‚K KT .t1; t2/ given in Proposition 2.4 D m;2 shows that the proposition holds in this case. For general n, similarly as in Section 2.3, we put

− n+1 n+1 − n+1 n+1 2 2 2 2

K L0 ; [ D D n/2 −n/2 n/2 −n/2 where we mean by a thick line 2 parallel copies of the line. For example, for n 4, it D is isotopic to

: D

By calculating its Kontsevich invariant in a similar way as in Section 2.3, it is shown 1 that the highest-degree part of Z.K L0/ is given by „ [ − t n−1 t 1 − Â Ã t n 1 1 t −1 −1 1 t − − t t Á − t 1 n t n 1 C 2 t n 1.t 1/ n 1 t .t 1/ t 1 Â n 1 1 t n 1 t 1n Ã t 2 1 n 1 1 n 1 C C t t ; Á C 4 C t 1 t 1 where the ﬁrst and second equivalences are obtained from Lemma 5.3 and Lemma 5.9 respectively. Hence, by a similar calculation as in Section 2.3, the highest-degree part .2–loop/ KT of Z .Km;n/ is given by

n 1 n 1 1 1 t .t 1/ t 1 t .t t 2/ t t 2 * n 1 1 n 1 1n + C C ! t 2 1 t t 1 n 1 n 1 1 n C C t n 1 t 1 n t 1=2 t t ; C m C C : C4 D 4 n 1 1 n m=2 t 1 t 1 t t C Further, the highest-degree part of the 2–loop polynomial ‚ KT .t ; t / is presented, Km;n 1 2 modulo the equivalence (5), by

n 1 1 1 n 1 (29) 12m t1 .t1 t1 2/ t2 2 C 1C 1 n 1 1 n n 1 n 1 1 n 1 n .t1 t 2/.t t /.t t t t / : 4 C 1 2 C 2 1 2 C 1 2

Geometry & Topology, Volume 11 (2007) 1424 Tomotada Ohtsuki

As its reduction, the highest-degree part of the reduced 2–loop polynomial ‚ KT .t/ y Km;n is presented by 2m.t 2n 2 t 2 2n/; C whose degree is equal to 2n 2. This implies that the degree of (29) is at least 2n 1. Further, the degree of the symmetrization of (29) is at most 2n 1. Therefore, the degree of ‚ KT .t1; t2/ is equal to 2n 1, as required. Km;n

Proposition 4.15 The t –degree of the 2–loop polynomial ‚ C .t ; t / of the gen- 1 Km;n 1 2 eralized Conway knot KC is equal to 4n 2. m;n

Proof For n 2, a concrete presentation of ‚K KT .t1; t2/ given in Proposition 2.5 D m;2 shows that the proposition holds in this case. For general n, we put

− n+1 − 2 n/2

K L0 : [ D n+1 n/2 2

By calculating its Kontsevich invariant similarly to the proof of Proposition 4.14 follow- 1 ing arguments in Section 2.4, it is shown that the highest-degree part of Z.K L0/ [ is given by „

− t n−1 t 1 ! n−1 − t 1 t t 1 −1 1 t − − t Á − t 1 n t 1 n C 2 t n 1.t 1/

−1 t − 1−n 1−n t .Á/ t t t n 1.t 1/ ! t n 1.t 1/ t n 1.1 t 1/ 1 ; t 1 Á C t 1 n.t 1/ where the ﬁrst and second equivalences are obtained from Lemma 5.3 and Lemma 5.10 respectively. Hence, the highest-degree part of 1Z.K L/ is given by L [ 1=2m n 1 n 1 1 t .t 1/ t .1 t / : t 1 .2–loop/ C Further, the highest-degree part of Z .Km/ is given by

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1425

n 1 n 1 1 2 1Â t .t 1/ t .1 t / Ã ; m=2 2 t 1 t2n 2.t t 1 2/ tn 1.t t 1 2/ C C 3 1 t2 2n.t t 1 2/ t1 n.t t 1 2/ m C C D 2 t t 1 2 C t t 1 2 C C t3n 3.t t 1 2/ t t 1 2 C C ! t t 1 2 1 t t 1 2 C C : t t 1 2 2 t t 1 2 C C

Hence, the highest-degree part of the 2–loop polynomial ‚ C .t ; t / is presented, Km 1 2 modulo the equivalence (5), by

3 1 1 1 1 12m .t1 t 2/.t2 t 2/.t1t2 t t 2/ C 1 C 2 C 1 2 1 t 2n 2t 2 2n t n 1t 1 n t 3n 3 1 : 2 1 2 C 1 2 1 2 Further, its symmetrization is presented by m3.t n 1t 1 n t 1 nt n 1 2/.t 2n 2t n 1 t 2 2nt 1 n 2/ 1 2 C 1 2 1 2 C 1 2 n 1 2n 2 1 n 2 2n 1 1 1 1 .t t t t 2/.t1 t 2/.t2 t 2/.t1t2 t t 2/; 1 2 C 1 2 C 1 C 2 C 1 2 whose t1 –degree is equal to 4n 2. Therefore, the degree of ‚ C .t1; t2/ is equal to Km 4n 2, as required. 4.3 Clasper surgery formulas

In this section, we show surgery formulas for the 2–loop polynomial under surgery along some types of graph claspers, as consequences of calculations of the previous section. A graph clasper is an embedded graph in a knot complement, such as shown in Lemma 4.16, deﬁned by

; ; D D D where the right-hand side of the ﬁrst formula implies the result of surgery along the Hopf link. The embedded graph in the ﬁrst picture is called a clasper, and a circle at an end of a clasper is called a leaf; see[12] for details of claspers.

Geometry & Topology, Volume 11 (2007) 1426 Tomotada Ohtsuki

As a particular case of a modiﬁcation of results in[20], we obtain the following lemma; see also[33] for calculations of the Kontsevich invariant for graph claspers.

Lemma 4.16 We have that 0 1 Â Ã x Z B C Z ; @ A x y Á x y x y y x y x y x y ! ! x y Z Z : Á z w z w z w z w

Sketch proof We show a sketch proof of the lemma; for details see[33]. Introducing a white trivalent vertex by

; D the left-hand side of the second formula of the lemma is equal to the Kontsevich invariant of x y x y

(30) ; D z w z w because, when we break one trivalent vertex of a graph clasper, the whole of the graph clasper vanishes. By Lemma 4.20 below, The Kontsevich invariant of a white vertex is presented by 0 1 Â Ã Z B C Z : @ A x y z .2/ Á x y z x y z

Hence, by replacing each clasper in (30) with a Hopf link, the Kontsevich invariant of (30) is obtained from x y

−1 −1

z w

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1427 by the Aarhus integral, which gives the surgery along the Hopf links. This gives the right-hand side of the second formula of the lemma.

The ﬁrst formula of the lemma is obtained in the same way.

Let F be a Seifert surface of a knot K. The Seifert form H1.F/ H1.F/ ޒ is ˝ ! deﬁned by taking a b to the linking number of a and b , where b denotes the ˝ C C puss-off of b in the normal direction of F . It is presented by a Seifert matrix, ﬁxing a basis of H1.F/. We denote by ex; ey the vectors presenting cohomology classes 1 T 1=2 1=2 T 1 x; y H .F/ for the basis. The scaler ex .t V t V / ey depends only on 2 1 the Seifert form and x; y H .F/, independently of the choice of a basis of H1.F/. 2 1=2 1=2 T The Alexander polynomial of the knot is given by K .t/ det .t V t V /. D A leaf of a clasper in the complement of a Seifert surface F of a knot is associated with a cohomology class in H 1.F/ counting cycles as

x :

The following two propositions can alternatively be obtained from a surgery formula in[19].

Proposition 4.17 Consider a graph clasper of the form in the following formula, embedded in the complement of a Seifert surface of a knot K. Let x, y be cohomology classes in H 1.F/ associated with the leaves of the graph clasper. Then, the change of the 2–loop polynomial of the knot by surgery along the graph clasper is presented, modulo the equivalence (5), by ! ! ‚ ‚ 12 Fxy.t1/ K .t2/ K .t3/; where Fxy.t/ 1=2 1=2 T 1=2 1=2 T 1 .t t / ex t V t V ey: K .t/ D

Proof Since the formula of the lemma is independent of a choice of a Seifert surface F and a basis of H1.F/, it is sufﬁcient to show the proposition for a particular choice of them. We assume that V is a Seifert matrix for a natural basis of H1.F/ of a natural Seifert surface F of KT in (27). We can also assume, without loss of generality, that

Geometry & Topology, Volume 11 (2007) 1428 Tomotada Ohtsuki

x and y are cohomology classes given by components Xi and Xj of T in (27). By Proposition 4.3, F .t/=.t/ ! ! x xy Z.2–loop/ Z.2–loop/ ˛; ; D D y where ˛ is given in Proposition 4.1, and Fxy.t/=.t/ given above is equal to the one given in Lemma 4.2. The corresponding 2–loop polynomial gives the required formula.

Proposition 4.18 Consider a graph clasper of the form in the following formula, embedded in the complement of a Seifert surface of a knot K. Let x, y , z, w be cohomology classes in H 1.F/ associated with the leaves of the graph clasper as shown at the leaves. Then, the change of the 2–loop polynomial of the knot by surgery along the graph clasper is presented, modulo the equivalence (5), by

x y ! ! ‚ ‚ z w 12 Fxy.t1/ Fzw.t2/ Fxw.t1/ Fzy.t2/ 1 Fxz.t1/ Fzx.t1/ Fyw.t2/ Fwy.t2/ K .t3/; C 2 where FXY .t/ is the one given in Proposition 4.17.

Proof In the same way as the proof of Proposition 4.17, 0 1 0 1 x y .2–loop/ B C .2–loop/ B C Z B C Z B C ˛; @ A @ A D z w

Fxy .t/=.t/ Fxw.t/=.t/ .Fxz.t/ Fzx.t//=.t/ 1 : D C 2 Fzw.t/=.t/ Fzy .t/=.t/ .Fyw.t/ Fwy .t//=.t/ The corresponding 2–loop polynomial gives the required formula.

The following proposition can alternatively be obtained by using a surgery formula in [26].

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1429

Proposition 4.19 Let T be a tangle as in (27), and let T C be a tangle obtained from T by surgery along a graph clasper C as

T C : D

Xa Xb Xc Suppose that

ni =2 kij Xi Xl 1 Y Y X Á Z.T / .2/ 1 ijl : Á t X X t X X C 1 i 2g i i 1 i

1Z.T C / 1Z.T / n =2 k X X X X X X Y i Y ij Â a c Â 1 a c X i l Ã 1 ijl Á t X X t X X 2 C 1 i 2g i i 1 i

Lemma 4.20 We have that

Geometry & Topology, Volume 11 (2007) 1430 Tomotada Ohtsuki

Â Ã Â Ã Z Z x y z x y z x z x z x x y y z z 1Â Ã : Á C x y z 2 y y C y z C x z C x y x y z

Proof The left-hand side of the required formula is presented by

0 1 0 1 B C B C (31) Z B C Z B C : @ A @ A

x y z x y z

We calculate the difference of the part surrounded by the box. We have that

− ! 1 x x z z +1 −1 Â 1 1 Ã Z +1 1 ; − Á 1 Á C 2 y z C 2 x y x y z x y z x y z where the second equivalence is obtained from the following formula,

−1 x x x z z Â 1 1 Ã +1 1 ; z y Á C 2 y z C 2 x y which is obtained by Lemma 5.6. Hence,

! ! Z Z

x y z x z x z x x z z 1Â Ã : Á C 2 y y C y z C x y x y z x y z

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1431

Therefore, (31) is calculated as follows,

Â x z x z x x z z Ã 1/2 1 − x y z C 2 y y C y z C x y

x y z

x z x z x x y y z z 1Â Ã : Á x y z C 2 y y C y z C x z C x y

This gives the right-hand side of the required formula.

Example 4.21 Using Proposition 4.19, we calculate the 2–loop polynomial of the KT Kinoshita–Terasaka knot Km again, to verify Proposition 2.4, as follows. The Kinoshita–Terasaka knot is presented by

m m m

KKT m D D D

m m

: D D

It is isotopic to the boundary of the ribbon graph given by the following knotted trivalent graph:

Geometry & Topology, Volume 11 (2007) 1432 Tomotada Ohtsuki

Hence, the Kinoshita–Terasaka knot is isotopic to KT C given in (27) for

C m T m D D

XXXX1 2 3 4 where T denotes the tangle in the middle picture ignoring the graph clasper C , and T C denotes the tangle obtained from T by surgery along C , noting that KT is isotopic to the trivial knot. We have that

1/2 m/2 −1 Z.T C / Z.T / Á XXXX1 2 3 4 X X X X X X 1Â X1 X3 X1 X3 1 1 2 2 3 3 Ã : C 2 C X2 X2 X2 X3 X1 X3 X1 X2 XXXX1 2 3 4 Hence,

1=2 1 m=2 Â X1 X3 X1 X3 X1 X3 X1 X1Ã 1 C 1 1 1 Z.T / Z.T / : Á X X X X X X C 2 2 2 2 2 3 4 4 X2 X2 X2 X2 X3 By Proposition 4.19,

.2–loop/ KT .2–loop/ C .2–loop/ ˝ ˛ Z .K / Z .K / Z .KT / ˛; ˇ ; m D T D where KT .t/ K C .t/ 1, and D T D Y Â Xj Xi Zj Xi Zj Zi Ã ˛ ; D t 1 i;j 4 FXi Xj.t/=2 FXi Zj.t/ FZi Zj.t/=2 Ä Ä X X X X Z Z Z X X 1 1 3 Â 1 3 X Â i i i ÃÃ 1 1 1 ˇ t ; D 2 C C t 2 X2 X2 i 1;3 Zi 1 Zi 1Zi 1 X2 X3 D C C C and FPQ.t/’s are given by

0 1 1 t 2 t 2 1 2 C 2 1 2 t 1 m.t 3t 3 t / t 2t 1 B m.t 4t 6 4t t / C C C T C C 1 C F .t/ e B 1 t 0 0 0 C ej ; Xi Xj i B C C D @ m. t 3 3t 1 t 2/ 0 m. t 2 t 1/ t 1 A C C C 1 2t 1 t 2 0 1 t 1 0 C C

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1433

0 1 t 1 m.t 2 3t 3 t 1/ t 1 m.t 2 2t 1/1 C C C C T B 0 1 0 0 C FXi Zj .t/ ei B 1 C ej ; D 0 m. t 2 t / 1 m. t 1/ @ C C A 0 1 t 1 0 1 C 0 0 0 0 0 1 1 T B 0 1 m. t 2 t / 1 m. t 1/C FZ Z .t/ e B C C C C ej : i j D i 0 1 0 0 @ A 0 m.1 t 1/ 0 m We have that

FX1X2 .t/ t 1 X1 X3 X1 X3 1 1 m 1 FX X .t/ t 1 ˛; 2 1 ; 2 D 2 D 2 1 X X FX3X3 .t/ t t 2 2 2 C X1 X3 Z1 Z1 1 t 1 ˛; 0; 2 D X2 Z2 FX1Z1 .t/ FX1Z1 .t/ X1 X3 Z1 1 1 1 1 FX2Z2 .t/ .t 1/FX2Z2 .t/ ˛; t 1 2 D 2 .t 1 1/F .t/ C 2 F .t/ X2 Z2 Z2 X3Z2 X3Z2

m m t 1 1 ; 2 1 1 2 1 D .t 1/.t t 2/ t t 2 C C Z Z 1 X1 X3 3 3 ˛; t 1 0; 2 D X2 Z4 F .t/ 1 X X Z3 X1X2 .t 1/FZ4Z4 .t/ t 1 t 1 1 1 3 1 m FX3Z3 .t/ ˛; t 1 ; 2 D 2 D 2 X2 Z4 Z4 F .t/ X X X1X3 F .t/ F .t/ 1 1 1 1 1 X2X1 X1X3 ˛; 2 D 2 FX X .t/ 2 X2 X3 2 1 2 1 2 2FX1X3 .t/ FX3X1 .t/ 2t 5t 3 t t 1 m C C : 2 2 1 D FX2X1 .t/ D t 1 t 1 m 1 m .2–loop/ KT t 1 Hence, Z .Km / 2 1 2 1 1 D t t 2 C .t 1/.t t 2/ C C t 1 2t2 5t 3 t 1 t 2 C C m t 1 1 m m : 2 t t 1 2 2 t t 1 C 2 t 1 1 C

Geometry & Topology, Volume 11 (2007) 1434 Tomotada Ohtsuki

Therefore, by deﬁnition, the 2–loop polynomial of the Kinoshita–Terasaka knot is presented by

‚ KT .t1; t2/ m .2T1;0 2T2;0 2T2;1 T3;1/; Km D C as we showed in Proposition 2.4.

5 Calculation of Gaussian diagrams

In this section, we develop methodology to calculate Gaussian diagrams. We prove basic formulas in Section 5.1, calculate the PBW isomorphism for Gaussian diagrams in Section 5.2, and show further formulas to calculate the 2–loop polynomial for knots of genus 1 and for knots of any genus in Section 5.3 and Section 5.4 respectively.

5.1 Basic formulas for Gaussian diagrams

In this section, we show basic formulas to calculate Gaussian diagrams. Some of the formulas are useful when we move an exponential chord beyond other chords.

Recall that a box over parallel chords denotes the symmetrizer of the chords as in (3).

Lemma 5.1

n 1 .n 1/.n 2/ : Á 2 C 6 n lines

In particular,

n 1 : Á.2/ 2 n lines

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1435

Proof When the left-hand side of the following formula is a part of a diagram, we can calculate the part as follows,

1 1 1 1 1 D n C n C n C n C C n

n 1 n 2 n 3 1 ; D n n n n where n is the number of chords shown in the left-hand side. Diagrams in the right-hand side are further calculated as follows,

; D D

2 ; D D Á

: :

.n 2/ : D D Á

Hence, we obtain the ﬁrst formula of the lemma. The second formula of the lemma is a reduction of the ﬁrst formula.

Lemma 5.2 For a power series f ,

f f f 1 f 1 x y Á 1 x y f ; y y f f x Á x C 8 12 x y

1 1 Á 1 : Á 8 C 12

Geometry & Topology, Volume 11 (2007) 1436 Tomotada Ohtsuki

Proof The ﬁrst formula of the lemma is easily obtained from the second formula. The second formula is reduced to its expansion,

n.n 1/ n.n 1/.n 2/ (32) ; Á 8 C 12 where the diagrams are of degree n. We show this formula by induction on n. For simplicity, we calculate the case n 4; the general case can be calculated in the D same way. By Lemma 5.1, we have that

3 (33) : Á 2 C

The last diagram vanishes because

1 0: Á D 2 Á

By substituting the following relation

1 1 1 Á.2/ 2 2 2 into the second last diagram of (33), we have that

3 3 Á 4 C 2

3 2 ; Á 2 C where the second equivalence is obtained from the assumption of induction,

3 1 : Á 4 C 2

Hence, we obtain (32) for n 4. D

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1437

Lemma 5.3 For a power series f ,

f f f Â 1 f 1 x y Ã 1 f : xf y f x y Á x y 4 C 6 x y

Proof By Lemma 5.2,

f f Â 1 f 1 f Ã 1 f : Á 8 f C 12 f

By applying the antipode on the right string and replacing f with f , we have that f f Â 1 1 Ã 1 : Á C 8 12

Since the diagrams in the left-hand sides of the above two formulas are equal, we obtain the required formula.

Lemma 5.4 For power series f and g,

y y f f f y f y f Â y Ã − 1 f 1 g 1/2 g x x g 1 g g z z Á g z C 6 C 6 x z z x

f y f y f y Â x x Ã − f 1 1 1 1 1/2 f g g 1 g g : g z x x Á C 6 C 6 C 4 z C 4 z x

Proof The second equivalence of the lemma is easily obtained by using the relation,

f f 1 f : Á.2/ 2

The ﬁrst equivalence is rewritten as Lemma 5.5 below.

Lemma 5.5 For power series f and g,

f y f y y f Â Ã −1/2 1 1 g g 1 : g z 6 6 x z Á C C x z

Geometry & Topology, Volume 11 (2007) 1438 Tomotada Ohtsuki

Proof For simplicity, we omit f , g, x, y , z in diagrams of the proof. By Lemma 5.1,

n m 1 .n m 1/.n m 2/ C C C ; Á 2 C 6 where n (resp. m) is the number of chords whose right ends are labeled by y (resp. z) in the diagram of the left-hand side. The last two terms are calculated as follows,

n m 1 m C ; 2 D 2

.n m 1/.n m 2/ m.n m 2/ C C C 6 D 6

m.n 1/ m.m 1/ : D 6 C 6

m m.n 1/ m.m 1/ Hence, : Á 2 6 C 6

This formula implies the mth part of the expansion of the following formula,

1 n 1 1 ; Á 2 6 C 6 where n is the number of chords whose right ends are labeled by y (relatively upward ends) in the diagram of the left-hand side. From this formula, we obtain the following formula by induction on n,

n n.n 1/ n n.n 1/ : Á 2 C 6 C 6 C 8

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1439

This formula implies the nth part of the expansion of the upper double lines of the following formula,

Â 1 1 1 Ã 1 1 : Á C 6 C 6 C 8 2

This implies the required formula.

Lemma 5.6 For power series f and g,

y f y f y f y f Â y f 1 x f 1 g −1 1 x g Á g C 2 g C 2 g z z g z z z x x f y f y 1 x 1 x Ã f g g g : x x C 2 z C 2 z

Proof The lemma is easily obtained from the following formula,

y f y f y f Â Ã −1 1 1 g g 1 : g z Á z C 2 C 2 x x z

We show this formula, omitting f; g; x; y; z for simplicity.

By applying Lemma 5.5 twice, we have that

−1/2 Â 1 1 Ã 1 C 6 C 6 Á D

Â 1 1 Ã 1/2 1 ; Á C 6 C 6 hence

Â 1 Ã 1 : Á Á 1/2 2

It follows that

Geometry & Topology, Volume 11 (2007) 1440 Tomotada Ohtsuki

−1/2 Â 1 Ã 1 Á −1/2 C 2

−1 Â 1 1 Ã 1 ; Á C 2 C 2 as required.

Lemma 5.7 For power series f and g, f+g f f+2g f+2g f f Â 1 y 1 x 1 x y f f g g 1/2 1 x y x y f g g g x y Á f 6 y 6 x 4 x y 1/2 g x y g f g 1 x y 1 x y 1 x y f g g f g f 4 x y C 4 x y C 4 x y

f g 1 x y 3 x y 1 f Ã g f : f g x g y C 4 x y 4 x y 4

As a corollary, the following formula is easily obtained from the lemma, f+g f −g f−g f Â f y x 1/2 1 f 1 g g 1 x y x f y f g g Á 1/2 C 12 y C 12 x x y g x y 1 1 1 Ã : 4 4 C 4

Proof of Lemma 5.7 For simplicity, we omit x, y in diagrams of the proof. By applying Lemma 5.4 to the left box of the second diagram of the following formula, we have that

f f f f+g Â 1 f 1 Ã − f f g (34) g 1/2 g 1 g D Á g C 6 C 6 g

f g Â 1 1 Ã 1 ; − f Á 1/2 g C 6 3

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1441 where the last equivalence is obtained from the following formula,

f f f f f g g g g f f g : g D g Á g f g

Here, the equivalence is obtained from the relation,

g g g g : Á.2/

Since

f f f g g g f g ; Á.2/ + we have that

f g f Â 1 f+g 1 f+g Ã g f f g Á g C 2 g C 2 f

f g Â 1 1 1 f Ã f : g Á g C 2 C 2 2

Hence, from (34), we have that

f f f+g Â 1 f 1 g 1 f g f g g Á −1/2 C 6 3 g 1 1 1 Ã : C 4 4 C 4

Geometry & Topology, Volume 11 (2007) 1442 Tomotada Ohtsuki

By applying Lemma 5.4 again, we have that

f f g f Â f f −1/2 1 1 f g 1 f g −1/2 g Á g C 6 g C 6 g f −1/2 g 1 f 1 g Ã g f C 4 g C 4 f f g Â 1 1 f 1 −1/2 Á g C 6 3 f −1/2 g 1 1 1 g Ã f : 4 C 4 C 2 g

Therefore,

f+2g f+2g f+g Â 1 1 1 f f 1 f f Á C 6 g C 6 g C 4 g

1 1 1 g g C 4 4 4 f

1 f 3 1 f Ã g : 4 f C 4 C 4 g

Hence, we obtain the formula of the lemma.

Lemma 5.8 The following formulas hold for a power series f and a scalar c, under the notation (13),

f Â c c 2c Ã f+c 1 ; c Á 2 C 2 3

f−c Â c c 2c Ã f 1 : c Á 2 C 2 3

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1443

Proof By Lemma 5.7,

f+c f f Â c f c f 1/2 f f c c 1 Á f 6 6 1/2 c c f c f Ã C 4 f 4

f+c Â c f f c f f c f f c c Ã 1 : Á 6 6 4 4 C 2

By removing the boxes by Lemma 5.2, we obtain the ﬁrst formula of the lemma. The second formula is obtained from the ﬁrst one by replacing f with f c.

Lemma 5.9 The following formula holds for a power series f , under the notation (13),

f Á 1 : −f Á C

Proof We have that

1 f f f f f D Á C C 2

Â 1 3 1 1 Ã 1 : Á C 2 2 C 2 2

On the other hand, by Lemma 5.5, we have that

f f f f f 1/2 −f

− − D Á f Á −f f Á f 1/2 f

Geometry & Topology, Volume 11 (2007) 1444 Tomotada Ohtsuki

f Â 1 1 1 1 Ã − 1 ; Á f C 2 2 2 C 2 where we obtain the last equivalence from Lemma 5.2 and the following formula,

f f Â Ã 1/2 1 1 f 1 f f f f 1 Á C 2 Á 2 C C 2 f

Â 1 1 Ã 1 f f : Á C 2 4

Hence, we obtain the required formula.

Lemma 5.10 For a power series f ,

f f f f f Â 1 1 f f 1 f 1 f f 1 f f Ã 1 f f : Á −f C2 2 f 2 f 2 C2 −f

Proof By Lemma 5.6,

f f 1 Á Â 1 Ã f 1 f f 1 f f : f Á f Á C 2 Á C 2 −f −f −f

By applying Lemma 5.6 again in another way,

f f f f f Â 1 f f 1 f 1 f f 1 f f Ã 1 −f f f Á f 2 2 C 2 2

f −f f f f Â 1 f f 1 f 1 f f 1 f f Ã 1 : Á f C 2 f C 2 f C 2 2 f

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1445

The lemma follows from the above formulas.

5.2 Calculation of the inverse image of Gaussian diagrams by the PBW isomorphism

In this section, we calculate the inverse image of an exponential chord by the PBW isomorphism. This result is used, in Section 2, to calculate the rational version of the Aarhus integral, which constructs the 2–loop polynomial of knots from their surgery presentations.

Recall the markings introduced in (13). For example, the following diagrams are calculated as follows,

f f 1Â Ã (35) ; D f D D 2 C D

1Â Ã (36) f f f : D D D 2 C D

Further, when the left-hand side of the following formula is a part of a diagram, the part is calculated as follows,

f 1Â Ã (37) f : D D 2 D

Lemma 5.11 Under the notation (13),

Â 1 1 1 f exp Á t C 2 6

1 1 1 1 Ã : C 4 2 C 3 C 3

Proof Since f 1Â Ã f ; D D 2 C D

Geometry & Topology, Volume 11 (2007) 1446 Tomotada Ohtsuki the right-hand side of the formula of the lemma is equivalent to f Â 1 1 1 1 1 C 2 C 8 2 C 2 1 1 1 1 1 Ã : 6 C 4 2 C 3 C 3 Hence, the lemma is reduced to the following formula,

1 Â1 1 (38) f f f Á C 2 f f C 8 2

1 1 1 1 1 1 Ã : C 2 6 C 4 2 C 3 C 3 The second term of the right-hand side is calculated as follows,

1 1 1 1 2 Á 2 f 2 Á 2

1 Â 1 1 Ã ; Á 2 f C 4 C 2 where the last equivalence is obtained by Lemma 5.12, and the second last diagram of the ﬁrst line vanishes because

: D D

Here, the ﬁrst equality is derived from the AS relation and the second equality is derived from the fact that the marking can move over by the IHX relation. The third term of the right-hand side of (38) is similarly calculated as follows,

Á f C

Â 1 Ã : Á f C C 2

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1447

Hence, from (38), the lemma is reduced to the following formula,

1 Â1 1 (39) f f f Á 2 f C f C 8 2

1 1 1 1 1 1 Ã : C 2 C 6 4 C 2 3 3

We show the nth part of the expansion of this formula by induction on n. In the following of this proof, for simplicity, we calculate the 4th part of the expansion of (39); the general case can be calculated in the same way. By Lemma 5.1, we have that

7 (40) f f f f f f f f f f f f 7 f f f f : Á 2 C

The last term is calculated as follows,

f f f f (41) 7 7 f f 6 f ; D f D f C f where the second last diagram is obtained by using the relation (37); this diagram vanishes as follows,

f f f 0: D f Á D

The last term of (41) is calculated as follows,

f f f 6 f f 4 D f C f C f

Â f Ã f f f 4 : Á C The ﬁrst term of the last line vanishes by (35) and (36). Hence, from (41), the last term of (40) is written

(42) 7 4 f : D

Geometry & Topology, Volume 11 (2007) 1448 Tomotada Ohtsuki

The second last term of (40) is calculated as follows,

f f 7 7 f 1 f (43) f f f f f f 3 f : 2 D 2 f D 2 f f

By applying Lemma 5.1 twice to the last term, we have that

f 15 f 3 3 f f D f C 2 f

f f 15 3 f 6 f : D f C f C 2

The last two terms are calculated respectively as follows,

f 6 6 f . 6/ ; D f Á

f f 15 3 f 3 f 6 f f 6 : 2 D 2 f C f Á 2 C

Further,

1 Â Ã 1 f : D f 2 f C f C f D 2

Hence, from (43), the second last term of (40) is written

7 1 f Â Ã (44) 3 f 6 6 : 2 D 2 f C

We calculate the ﬁrst term of the right-hand side of (40) by using Lemma 5.1 as follows,

(45) f f f f f f f f 3 f f f f 5 f f f f : Á C

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1449

The last term is calculated as follows,

f f f f 5 f f f f 5 f f 5 f f f f 4 f D f D D C

f f f 4 : Á C f

The second last term of (45) is calculated as follows,

f f f f 3 f f f f 3 f f 3 f f 3 f f 6 f f ; D f D Á C where the last equivalence is derived from Lemma 5.1. The last two terms are calculated as follows,

f f 6 6 f 6 ; D Á f Â f f Ã 3 3 f 12 12 Á f C C f Â Ã 3 3 f 12 12 . / ; Á f C C 2 where the last equivalence is obtained from the relation,

f f : C C D D C D

Hence, from (45), the ﬁrst term of the right-hand side of (40) is written

Â 3 Ã f f f f f f f f 3 Á C 2

Â Ã 4 6 12 12 : C C

Geometry & Topology, Volume 11 (2007) 1450 Tomotada Ohtsuki

Therefore, from (42), (44), and the above formula, (40) is rewritten

1 f f f f f f f f f f 6 f Á 2 f C f

Â 3 Ã Â Ã f f f 6 12 : C 2 C

The last two diagrams of the ﬁrst line are calculated by (47) and (48) as follows,

1 1 1 3 f f f f f f 3 2 D 2 Á 2 C 4

6 6 6 6 12 D f f Á f f C

Hence,

1 1 1 1 (46) 24 Á 24 48 C 4

Â 1 1 1 Ã C 24 16 C 32 Â 3 1 1 Ã 1 : C 8 4 2 C 2

By the assumption of induction,

1 1 1 f f f f f f f f 6 Á 6 4 C f

Â1 1 1 1 Ã C 8 C 6 4 C 2 Â 1 1 1 Ã : C 2 3 3

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1451

Substituting this formula into the ﬁrst term of the right-hand side of (46), we calculate the term as follows,

1 1 f f f f 1 f 1 f f f f f f f 24 Á 24 16 C 4 f

Â Ã f f 1 1 1 1 C 32 C 24 16 C 8 Â Ã f 1 1 1 1 C 8 12 12 4

1 1 1 f f f Á 24 16 C 4 f f

Â 1 1 1 1 Ã C 32 C 24 16 C 4 Â 1 1 1 1 Ã ; C 8 12 3 2 where the last equivalence is obtained by using the following relations,

2 ; f D C D

: f D C

Therefore, from (46),

1 1 1 1 f f f f 24 Á 24 12 C 2

1 Â1 1 1 1 Ã C 2 8 C 6 4 C 2 Â 1 1 1 Ã 1 : C 2 3 3 C 2

This is exactly the 4th part of the expansion of (39), as required.

Geometry & Topology, Volume 11 (2007) 1452 Tomotada Ohtsuki

Lemma 5.12 Under the notation (13),

1 f f : Á.2/ 2 f C f

In particular, as the second and third parts of the expansion of the formula of the lemma,

1 f f 1 f f 1 (47) f ; 2 Á.2/ 2 2 C

1 f f f 1 f f f 1 (48) f f : 6 Á.2/ 6 4 C f

Proof of Lemma 5.12 The formula of the lemma is a reduction of (39). In fact, it can be proved in the same way as the proof of (39), but the proof is far easier than the proof of (39), which we show in the proof of Lemma 5.11. A detailed proof is left to the reader.

Lemma 5.13 Under the notation (13),

! 1 1 1 1 f 1 .Á~/ C 12 4 C 12

Proof By Lemma 5.11,

Â 1 1 1 1 1 f 1 Á C 2 C 8 2 C 2 1 1 1 1 1 Ã : 6 C 4 2 C 3 C 3

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1453

The required formula is obtained from this formula by using the following link relations,

Â Ã 0 2 ; .Á~/ D Â Ã 0 2 2 ; .Á~/ D Â Ã 0 2 ; .Á~/ D Â Ã 0 2 : .Á~/ D

Lemma 5.14 Under the notation (13),

g y g x y Â x x 1 x 1 g exp x x y x f x D t 2 x C x x

x x x x 1 1 1 1 g 1 g 1 x x y x y 6 C 4 2 C 6 C 2 C 3 x x x x x

g x x g g y Ã 1 x 1 g 1 x y 1 x y 1 x y x x ; g g x C 3 x C 3 x 4 x y C 6 x y 2 x for power series f and g. In particular, when f is a scalar c,

g y g x x y g Â 2 x y Ã 1 x c c g c 1 x y : x c c g Á C 12 C 6 12 x y x x x

Proof The second formula of the lemma is easily obtained from the ﬁrst formula. In the following of this proof, we show the ﬁrst formula.

Geometry & Topology, Volume 11 (2007) 1454 Tomotada Ohtsuki

In a similar way as the proof of Lemma 5.2, we can show that

m lines

Â1 2 Ã

Á C f 3 C C 3 f f f

Âm 1 m 1 Ã : C f 2 3

By this formula we can show the mth part of the expansion of the following formula by induction on n,

Â 1 f f f Á f C 6

2 1 1 1 Ã : C C 3 C 4 6 C 2

Further, applying (38) to the ﬁrst diagram of the right-hand side, we have that

1 Á 2 f C f

Â 1 1 1 1 1 C 8 C 2 2 C 6 4

1 1 1 1 C 2 3 3 6 C

2 1 1 1 Ã C 3 C 4 6 C 2

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1455

1 Â 1 f Á f 2 f C f C 8

1 1 1 1 1 1 C 2 2 C 6 4 C 2 3

1 1 1 1 1 3 6 3 C 4 6

1 1 1 1 Ã ; 2 2 2 C 2 C where the second equivalence is obtained from the following relations,

Â Ã ; f f Á f Á

1 Â Ã ; f Á 2 f Á

1 Â 1 Ã : f Á 2 f Á 2

Therefore, we obtain the ﬁrst formula of the lemma.

5.3 Formulas for Gaussian diagrams used in Section 3.4

In this section, we show formulas for Gaussian diagrams which are used in Section 3.4 to calculate the 2–loop polynomial for knots of genus 1.

Geometry & Topology, Volume 11 (2007) 1456 Tomotada Ohtsuki

Lemma 5.15 For power series f , g and a scalar c,

x c c x y 2 y y x f c c f c 1 f x x x f y 1 x g Á 1/2 C 12 C 6 C 6 z g x g x x z y x g z f y f c c x g c x z x y f x g z x g 12 x 12 6 z z f y f y ! 1 f y 1 g x x C 12 g C 12 g z z z x c

y 2 ! x f c c c 1 x 1 : .~Áx / x g C 12 C 12 C 12 C 24 z

Proof For simplicity, we omit f , g, c in formulas of the proof.

In a similar way as in the proof of Lemma 5.14, we have that

k k.k 1/ ; Á 6 C 12 where k is the number of horizontal chords in the diagram of the left-hand side. By splitting the chords into two parts, we have that

n m Á 6 6

n.n 1/ m.m 1/ nm ; C 12 C 12 C 6

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1457 where n (resp. m) is the number of horizontal chords whose right ends are labeled by y (resp. z). Hence,

.1 ˛/; Á C where

1 1 1 1 1 ˛ : D 6 6 C 12 C 12 C 6

Therefore, by Lemma 5.11 and Lemma 5.16,

Â 1 1 1 Ã 1 ˛ Á −1/2 C 12 C 6 3

It follows that

Â 1 1 1 Ã 1 ˛ Á 1/2 C 12 6 C 3

Â 1 1 1 Ã 1 ˛ : Á 1/2 C 12 C 12 C 12

Hence, we obtain the ﬁrst equivalence of the required formula.

The second equivalence of the required formula is obtained from the following link relations,

0 ; .Á~/ D

Â Ã 0 ; .Á~/ D

Geometry & Topology, Volume 11 (2007) 1458 Tomotada Ohtsuki

Â Ã 0 ; .Á~/ D C

Â Ã 0 : .Á~/ D C

Lemma 5.16 For power series f and g,

f y f y f y f y ! g y 1 x f 1 g z x g 1 z : − f y g g z Á 1/2 C 6 3 x g z z x z

Proof By Lemma 5.5, Â 1 1 1 Ã 1 1 Á C 6 C 6 C 8 2 Â 1 1 1 Ã 1 1 : Á C 6 3 C 8 2 This implies the required formula.

Lemma 5.17 x y x y z − w t 1 Â t −1 z w z t w 1 −1 z w 1 1 1 z;w z −1/2 1 z − w t ~ w t 1 t . Áz;w/ z t C 8 C 12 z w 12 z w − w z w z 1/2 t w t t+1 1 z w 1 z w 1 z t z w t t 4 z w C 4 z w C 12 z

x x y y 1 t+1 w 1 1 Ã z w : t t −1 − C 12 w C 24 z wC 24 z t 1 w

Proof By Lemma 5.2 and Lemma 5.7,

−1 −1 Â 1 1 1 1 t Ã 1 t t Á t C 8 C 8 C 12 12 t

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1459

Â t − t t −1 1 1 1 1 1 − −1/2 t 1 t Á t C 8 C 12 12 −1/2 t 1 1 t 1 t+1 1 t+1 Ã t : 4 C 4 t C 12 t C 12 t

A left part of the ﬁrst diagram of the right-hand side is calculated as follows,

Â t t t Ã t −1 1 1 1 t −1 1 t t t −1/2 t Á C 8 C 8 C 4 −1/2 t − 1 t 1 1 t −1 t t : C 2 C 2

1 The map z takes diagrams of the right-hand side as follows,

Â Ã 1 1 z t −1 1 t −1 ; .~Áz / C 24

− t 1 Â t − Ã 1 t 1 1 1 t z t t −1 t ; Á C 2 C 2 .~Áz /

t −1 Â − Ã 1 t 1 t 1 1 t z t t ; Á C 2 C 2 .~Áz / where the ﬁrst formula is obtained from Lemma 5.18 below, and the second and third formulas are obtained by calculating them directly. Hence,

− t 1 Â Ã 1 −1/2 1 z t 1 : .~Áz / −1/2 C 24 t

Geometry & Topology, Volume 11 (2007) 1460 Tomotada Ohtsuki

Further, in the same way, we have that

− t 1 t −1 Â Ã 1 − 1 1 1/2 −1/2 1 − − : z;w t t t 1 t 1 − .~Áz;w/ C 24 C 24 1/2 −1/2 t t

The required formula follows from the above formula and the ﬁrst formula of the proof.

Lemma 5.18 For power series f and g,

y y f f y f x Â f y Ã 1 1 x x g g 1 : ~ x g z . Áx / C 24 x z z

Proof By deﬁnition,

Â 1 Ã 1 (49) 1/2 1 : Á C 8 C 2

The ﬁrst diagram of the right-hand side is calculated, by Lemma 5.4, as follows,

Â 1 1 1 Ã 1 1 : Á C 8 C 6 C 6 2

Further, the last diagram of (49) is directly calculated as follows,

Â 1 1 1 Ã : Á C 2 2 2

Hence, (49) is rewritten

Â 1 1 Ã 1 : Á 12 12

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1461

Therefore, from the deﬁnition of ,

Â Ã 1/2 1 1 1 1 : Á C 12 C 12

The required formula follows from this formula by using the following link relations,

0 ; .Á~/ D

Â Ã 0 ; .Á~/ D C

Â Ã 0 : .Á~/ D C

Lemma 5.19

n/2 m/2 n/2 m/2

z w x x k y y x x y y Â 1 1 x y y k z t z w x x x y y 1 w t t−1 .Á~/ C8 z C 8 w z w − z t 1 w z z t/2 w z w z w z t w Ã w 1 1 z t t/2 w : z z w z w w C 4 4

Proof For simplicity, we omit n, m, k in some diagrams in the proof. We have that

n/2 m/2

x x k y y x y x y 0: − t 1 D .~Áz / .~Áx / .~Áy / D z w t−1 t−1 t−1 t−1 z t w w z w

In the same way, n/2 m/2

x x y y k x x y y 0: t− ~ z 1 w .Á/ z z t w w z

Geometry & Topology, Volume 11 (2007) 1462 Tomotada Ohtsuki

Similarly,

n/2 m/2

Â z t z w z t w Ã x x k y y w 0 x x y y z t w t ; − z z w .Á~/ t 1 D w C z w t t− z z 1 w w

t w w Â z Ã w z t w 0 t w : ~ − z w .Á/ t 1 D w C t

Further, in the same way,

Â z w z t w Ã 0 t ; .Á~/ C z w z w

z t z Â w Ã z z t w 0 z t : ~ w z .Á/ z C

The required formula follows from the above relations.

Lemma 5.20

z z z z y Â t −1 Ã t 0 : t t .Á~/ w w w w w

Proof The lemma is obtained from the following link relation,

0 : t−1 .Á~/ t

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1463

5.4 Formulas for Gaussian diagrams used in Section 4

In this section, we show formulas for Gaussian diagrams which are used in Section 4 to calculate the 2–loop polynomial for knots of any genus.

Lemma 5.21 For a scalar c and power series f1; f2; ; fn ,

c x c f f f y 1 y1 x 1 y Â i i Ã 1 x 1 Y 1/2 f f x fj x 2 y 2 2 x y2 Á yj 1 i

Proof For simplicity, we omit f1; f2; ; fn in diagrams of the proof. In a similar way as in the proof of Lemma 5.15, we have that

.1 ˛/; Y Y x Á x C

where we put Y y1 y2 yn and D C C C

1 1 1 x Y ˛ x Y : D 12 6 C 12 x Y

Geometry & Topology, Volume 11 (2007) 1464 Tomotada Ohtsuki

Therefore, by Lemma 5.22 below,

Y y2 x D y x n

yi yi y 1 Â Ã yi 1 yi 1 −1/2 y X x X Y 2 1 ˛ x # ; yj # x yj Á C C 6 3 y i

yi yi y 1 Â Ã yi 1 yi 1 1/2 y X x X x Y 2 1 ˛ # Y yj # x yj Á 6 C 3 y x i

x Â yi Ã Y x 1/2 x x Á x Y yj i

Lemma 5.22 For power series f1; f2; ; fn , y y f1 1 f1 1 y y fi i fi i f f fi y 2 y 2 y Â f yi Ã i 2 2 1 X i 1 X fj Y −1/2 x x f 1 f y # # j fj k j x f f 6 3 yj n Á n C y y x x i

Proof For simplicity, we omit x; f1; f2; ; fn . By Lemma 5.16,

y y y y 1 1 1 1 y y + y Â y Ã 1 2 3 1 1 1 y2 y + y y + y 2 3 −1/2 1 1 y2 y3 : 3 + D Á + C 6 y + y 3 y2 y3 y2 y3 2 3

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1465

By applying Lemma 5.16 again,

y 2 y2 y2 y y Â y Ã + 2 3 1 2 1 y2 y3 y y 3 −1/2 2 1 y3 : D Á C 6 y 3 y y3 3 3

From the above two formulas, we obtain the required formula for n 3. D The required formula for any n is obtained similarly by induction on n

Lemma 5.23 For a scalar c and power series f1; f2; ; fn ,

c x c x f f f y 2 1 y1 x 1 y Â i i Ã Â 1 x 1 Y 1/2 c f f x fj x 2 y 2 1 2 x y2 .~Áx / yj C 12 2 i

Proof The lemma is obtained from Lemma 5.21 by using the following link relations,

y1 x y x 1 x x x ; x Y .Á~/ x Y Z

x Â x Y Ã x x x Y Y ; x Y .Á~/ x Y

y1 y1 y x y 1 Â 1 y Ã x 1 x y x x 1 x ; Z .Á~/ Z C x Y Z Z

x y1 x y Â Ã x 1 Z x Z ; x Y .~/ Y Á C Z where we put Y y1 y2 yn and Z y2 yn . D C C C D C C

Geometry & Topology, Volume 11 (2007) 1466 Tomotada Ohtsuki

Lemma 5.24 For scalars a and c,

v c X u X v u a c a Â x x x X Ã 1 x x x c 1 x 1 : .Á~/ x 12 C 24 x z X z z

z z

Proof Similarly as in the proof of Lemma 5.15, we have that

c c x X XX X v Â c2 c c a Ã X X v v 1 : Á 12 6 C 12 x x 2 x u x a u x a u x x

Further, by modifying Lemma 5.23, we have that

z x z x x X c c Â 2 z z X v Ã 1 x c c 1 a X x X 1 : v .Á~/ v C 12 C 12 C 24 C 2 x x x x X x u x a u a u

We obtain the required formula from the above two formulas, using the following link relation, x z x c X x Â XX z X Ã x X v : x Á a u x x x x x x X

Lemma 5.25 For scalars n1; n2; k12 ,

n n2/2 1/2 n1/2 n2/2 xi xi Â Ân2 Ã 1 x1 k12 x x x x X i ni 1 1 k12 2 2 x x x x 1 2 .Á~/ x1 1 2 2 C 48 C 24 1 i 2 xi zi z z z z1 2 Ä Ä 1 2 x 3 x1 x1 x1 x2 x1 x2 2 1 k k k 3k n1k12 n2k12 Ã 12 12 12 12 C C : 12 C 24 C 24 C 24 x2 x2 z1 z1 z2 z2 x2 Proof By Lemma 5.2,

n1/2 n2/2 x1 Â k2 k3 x1 x1 Ã x1 k12 1 12 12 : Á x2 C 8 12 x2 x2 x2 z1 z2

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1467

By applying Lemma 5.15 to each solid circle of the ﬁrst diagram of the right-hand side, we obtain the required formula.

Lemma 5.26 For scalars ni; kij ; b,

z 1 z1 n n1/2 1/2

x1 x1 x1 x1 x1 x1 1 k12 k13 k12 k13 b n2/2 n3 ~ n2/2 x x n3/2 /2 .Á/ x2 2 3 x3 x2 x3 x x k23 2 x2 k23 x3 3 x 2 x3 z z z2 z3 2 3 x 1 x x Â Â 2 i i Ã k12k13 k12k23 k13k23 X n ni b C C 1 i C 2 C 48 C 24 1 i 3 xi zi x2 x3 x Ä Ä 2 i 3 xi xi xi xj xi xj Â3k nikij nj kij k k k Ã X ij C C ij ij ij C 24 12 C 24 C 24 1 i

Proof We obtain the terms of the right-hand side of the required formula labeled by at most 2 colors in the same form as in Lemma 5.25. It is sufﬁcient to calculate the terms labeled by 3 colors. In this proof we consider the equivalence modulo the diagrams with 2 trivalent vertices and at most 2 colors.

The diagram of the left-hand side of the required formula is equivalent to

n1/2

x1 Â x1 x1 Ã k12 k13 b.k12 k13/ b 1 C : n2/2 n3/2 2 C x2 x3

k23 x 2 x3

z2 z3

Geometry & Topology, Volume 11 (2007) 1468 Tomotada Ohtsuki

By applying Lemma 5.27 to the solid circle labeled by x1 , the ﬁrst diagram is taken 1 by x1 (modulo the equivalence) to

n1/2

x1 x1 x1 x x x 1 1 1 x2 x2 x3 x3 1 Â 2 2 Ã k12 b+K23 k13 k k13 bk Á k12k bk Á 1 12 12 13 13 ; 12 2 12 2 n2/2 n3/2 C C x1 x3C C x1 x2

k23 x 2 x3

z2 z3

i where we put K kij k =2. Further, the ﬁrst diagram is equivalent to jl D il

n1/2

x1 x1 x1 x1 x1 x1 Â x1 x1 x2 x2Ã k12 +K 1 k k12 k13 k12k13 Á k23 k12k13 Á b 23 13 1 C b b : 2 C 2 C 2 C 2 n2/2 n3/2 x2 x3 x1 x3

k23 x 2 x3

z2 z3

By applying Lemma 5.27 to the solid circle labeled by x2 , the lower part of the ﬁrst 1 diagram is taken by x2 (modulo the equivalence) to

x1 x1 x1 2 x1 x1 k12 1 2 k Â b+KK23 + 13 13 k k23 k k k ÁÁ 1 12 12 b 12 13 n2/2 x2 n3/2 C 12 C 2 C 2 x2 x2 x2 x3 x2 x x k 2 2 23 2 x3 x3 Ã x3 k12k23 k23 k12k13 ÁÁ z z b : 2 3 12 2 2 C C C x1 x2

Further, the right part of the ﬁrst diagram is equivalent to

x1 x1

k13 1 2 b+KK23 + 13 Â x2 x2 Ã n3/2 k23 k12.k13 k23/Á x2 1 b C : x2 2 2 k23 C x1 x3 x3 z3

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1469

By applying Lemma 5.27 to the solid circle labeled by x3 , the ﬁrst diagram is taken 1 by x3 (modulo the equivalence) to

x1 x1 Â 2 x1 x1 1 2 3 k13 k k23 k13 k12k13 k12k23 ÁÁ b+KK23 + 13+K12 13 n3/2 1 b C x3 x x 3 C 12 C 2 C 2 x2 3 x2 x3 x2 x3 x k x3 3 23 2 x2 x2 Ã k13k23 k23 k12k13 k12k23 ÁÁ z3 b C : 12 2 2 C C C x1 x3

By summing the diagrams in the right part of the above formulas, we obtain the terms of the required formula colored by 3 colors, completing the proof.

Lemma 5.27 For scalars a1; a2; a3; b; c,

c x c a a y x a1 b+ 2 3 2 a1 y1 1 y1 x x 2 a2 x a2 x y2 y2 ~ y3 . Áx / x a3 b y2 y3 y x y a3 3 1 y Â 2 2 3 a a y1 x c X aic X 1 j x 1 x yi C 12 C 12 C 24 y x 2 i 3 2 j 3 j Ä Ä Ä Ä y2 y2 2 y 2 Ã a a3 a b Á 2 a2a a b Á 2 2 x 3 3 x y3 : 12 2 y 12 2 C C 3 C C y3

Proof By deﬁnition,

c c a y2 1 y1 a Â 2 y Ã 1 y1 x 2 a b 2 y x 2 a2 1 y y : y2 3 b 2 Á a3 C 2 y3 C y y3 x 3 a3 y x 3

In a similar way as Lemma 5.21, we have that

y y ÂÂ 2 Ã Â i Ã aiaj X 1 b x 1 x x y Á 3 C 2 yj 1 i

C 2 y3 C 2 C 2

Geometry & Topology, Volume 11 (2007) 1470 Tomotada Ohtsuki

The error term between the required formula and a particular case of Lemma 5.23 is

x c y y ÂÂ 2 Ã Â b a a i Ã x a1 y x i j X x x 1 b 1 a2 y3 yj x y2 C 2 C 2 x a3 1 i

a1 y1 y x c ÂÂ 2 Ã a x 0 x 1 y1 b x y3 .Á~/ a2 D x y2 b y2 x y y 1 3 y1 y1 x a3 Â Ã y2 Ã y x x x 3 a1a2 a1a3 a1b : y2 y3 C C y3

Hence, from a particular case of Lemma 5.23 and the above mentioned error term, we obtain the required formula.

Lemma 5.28 For scalars ni; kij (with kij kji ), D

z2 z1 z2 z1 n x2 x1 2/2 n1/2 n n k12 2/2 x2 x1 1/2 x2 k12 x1 x2 x1 x Xi x 2 x1 x k24 2 x2 k24 x1 1 1 k Y ij l k23 k14 23 k14 .1 ˇ/; x x x x k13 .Á~/ 3 3 x k13 x 4 4 C 3 4 t Xj Xl x3 x4 1 i

Geometry & Topology, Volume 11 (2007) On the 2–loop polynomial of knots 1471

where we put .kij k kij k k k /=2 and ijl D il C jl C il jl

x x Â 2 i i Ã X n ni 12ˇ i D 4 C 2 1 i 4 xi zi Ä Ä x Â 2 i xi xi xi xj xi xj Ã X 3kij nikij nj kij kij kij C C k3 C 2 ij C 2 C 2 1 i

Proof We obtain the terms of the right-hand side of the required formula labeled by at most 3 colors in the same form as in Lemma 5.26. It is sufﬁcient to calculate the terms labeled by 4 colors, similarly as the proof of Lemma 5.26. In this proof we consider the equivalence modulo the diagrams with 2 trivalent vertices and at most 3 colors.

1 The diagram of the left-hand side of the required formula is taken by x1 (modulo the equivalence) to

z2 z1 n1/2 x2 x x1 k12 x1 1 n2/2 1 x1 K24 x1 x1 1 x2 x1 x2 x1 K x1 Â Â ÃÃ 23 x1 k k k x 12 13 14 1 k 1 ; k23 k 1 14 13 K34 6 C x3 x4C x3 x4 k24 k34 n n3/2 x3 x4 4/2

z3 z 4

Geometry & Topology, Volume 11 (2007) 1472 Tomotada Ohtsuki

i where we put Kjl kij kil =2 as in the proof of Lemma 5.26. The ﬁrst diagram is 1 D taken by x2 (modulo the equivalence) to

x2 x1 k12 x2 x1 n2/2 x2 x1 n1/2 x x x 1 x1 x2 x1 x2 2 2 x1 Â x2 x1 x2 x1Ã k13 k24 k21k23k24 1 + 2 2 1 KK1 + 2 KK23 13 K34 K34 24 14 1 x 2 x1 C 6 C k23 k14 x3 x4 x3 x4 k34 1 x2 x1 1 x2 x1 ! n x3 x4 n K k23 K k24 3/2 4/2 24 23 : z3 z 4 2 2 C x3 x4C x3 x4

1 Further, the ﬁrst diagram is taken by x3 (modulo the equivalence) to x x1 1 x2 x1 x2 x k x 2 13 2 x2 x1 x2 x1 x2 x1 k24 x Â Â Ã 1 κ123 2 3 1 3 1 k k k K k x K34+K24 K34+K14 31 32 34 34 23 2 1 2 1 x3 x3 K24+K14 k23 x3 x 3 x C 6 C C 2 x x3 1 x3 x4 x3 x4 x3 x4 3 k14 x3 x x3 k34 3 x4 x2 x1 x2 x1 n4 1 2 2 Ã n3/2 /2 .K K /k34 K k13 z3 z 4 23 C 13 34 : 2 2 C x3 x4 x3 x4 1 Furthermore, the ﬁrst diagram is taken by x4 (modulo the equivalence) to x2 x1 x2 x x3 k 1 24 κ Â Â x2 x1 x2 x1Ã 2 3 x2 x1 κ 124 x1 .K K /k x2 134 k41k42k43 14 x x 34 24 x 4 4x k14 1 C x κ 4 4 3 234 x x 6 2 4 4 C x3 x4C x3 x4 C x3 x4 x x 3 4 x k34 x4 4 n4/2 1 2 x2 x1 1 3 x2 x1 z 4 .K K /k34 . K K /k24 Ã 24 C 14 34 C 14 : 2 2 C x3 x4C x3 x4 By summing the diagrams in the right part of the above formulas, we obtain the terms of the required formula colored by 4 colors, completing the proof.

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Research Institute for Mathematical Sciences, Kyoto University Sakyo-ku, Kyoto, 606-8502, Japan [email protected] http://www.kurims.kyoto-u.ac.jp/~tomotada/

Proposed: Vaughan Jones Received: 4 December 2005 Seconded: Rob Kirby, Cameron Gordon Accepted: 20 May 2007

Geometry & Topology, Volume 11 (2007)