arXiv:math/0301312v1 [math.GT] 27 Jan 2003 n h rddPoincar´e polynomial graded the and npriua,tehmlg groups homology the particular, In r ikivrat.Teamo hsppri ogv nexampl an polynomial give the to by is separated paper are this which of links aim mutant The invariants. link are irddcancomplex chain bigraded ls fgae hi complex chain graded of class retdlinks oriented osqetyteinvariant the consequently group n[h .Koao sindt h diagram the to assigned Khovanov M. [Kh] In Introduction 1 nti eto ebifl ealtedfiiino ki equiv skein of definition the ( recall triple A briefly [Ka]. we in section this In equivalence Skein 2 polynomial. Jones the for relation skein the uhthat such xeti ml egbrod hr hyaerespectively are they where neighborhood, small a in except hoeia nlso)euvlnerlto ” relation equivalence inclusion) theoretical ento 1 Definition 3. 2. 1. , L L ihasenrl iia otesenrlto o h oe polynomial. Jones the for relation skein is the example to similar an defi rule be skein such cannot a homology of with Khovanov that existence shows it The because important homology. Khovanov different L C 0 0 Kh ∼ i,j hvnvHmlg n Conway and Homology Khovanov and nti ril,w rsn nes xml fmtn ik with links mutant of example easy an present we article, this In ∼ ∼ ( L ( D L L L ′ 0 ′ 0 ′ into ) tpa eri nvri¨tBsl aur 2003 January Universit¨at Basel, Wehrli, Stephan when )( L and and The ,q t, + . , L := ) L L L L + C ki equivalence skein − + − L , i and +1 and X ∼ ∼ i,j − ,j C L , L L ( L ∗ D t L + ′ − ′ , i ∗ ′ 0 Kh q 0 .H rvdta h oooyequivalence homotopy the that proved He ). ( foine ik scle a called is links oriented of ) r isotopic, are j imply imply D Mutation oss igaswihaemtal identical mutually are which diagrams possess dim C ,wt differential a with ), ∗ osntstsyasenrlto iia to similar relation skein a satisfy not does , Abstract ∗ ( Q D L L H ( H − + nydpnso h retdlink oriented the on depends only ) 1 i,j stemnml(ihrsett set- to respect (with minimal the is i,j ∼ ∼ ( D ( D L L cniee pt isomorphism) to up (considered ) − ′ + ′ ) ∼ , , ⊗ ntesto retdlinks oriented of set the on ” Q D ) ∈ fa retdlink oriented an of d Z Kh htmp h chain the maps that [ ,t t, n opoethat prove to and ki triple skein − 1 ossetwith consistent ,q q, , foriented of e lnegiven alence − 1 ned ] fthe if , L L a . ′ ′ ′ for any two skein triples (L+,L−,L0) and (L+,L−,L0). It is easy to see that such a minimal relation as postulated in the definition actually exists. The definition is motivated by the following: Assume we are given an invariant f of oriented links, such as the Jones polynomial, which takes values in an arbitrary ring R and satisfies a relation

αf(L+)+ βf(L−)+ γf(L0) = 0, ∗ where α, β ∈ R and γ ∈ R. Then f(L+) is determined by f(L0) and f(L−), and f(L−) is determined by f(L0) and f(L+). The minimality of ”∼” implies: Theorem 1 Let L and L′ be skein equivalent. Then f(L)= f(L′).

3 Conway mutation

The mutation of links was originally defined in [Co]. We will use the defini- tion given in [Mu]. In Figure 1, the rectangular boxes represent an oriented 2-tangle T . Let h1, h2 and h3 be the half-turns about the indicated axes.

T T T

h1 h2 h3

Figure 1: The half-turns h1, h2 and h3

Define three involutions ρ1, ρ2 and ρ3 on the set of oriented 2-tangles by ρ1T := h1(T ), ρ2T := −h2(T ) and ρ3T := −h3(T ) (where −h2(T ) and −h3(T ) are the oriented 2-tangles obtained from h2(T ) and h3(T ) by re-

TT1 2

Figure 2: The closure of the composition of T1 and T2

versing the orientations of all strings). For two oriented 2-tangles T1 and ∧ T2, denote by T1T2 the composition of T1 and T2 and by (T1T2) the closure of T1T2 (see Figure 2).

2 Definition 2 Two oriented links L and L′ are called Conway mutants if there are two oriented 2-tangles T1 and T2 such that for an involution ρi ′ ∧ (i = 1, 2, 3) the links L and L are respectively isotopic to (T1T2) and ∧ (T1ρiT2) .

Theorem 2 Let L and L′ be Conway mutants. Then L and L′ are skein equivalent.

Proof. The proof goes by induction on the number n of crossings of T2. For ′ n ≤ 1, T2 and ρiT2 are isotopic, whence L ∼ L . For n> 1, modify a crossing of T2 to obtain a skein triple of tangles (T+, T−, T0) (with either T+ = T2 or T− = T2, depending on whether the crossing is positive or negative). ′ ′ ′ Denote by (L+,L−,L0) and (L+,L−,L0) the skein triples corresponding to ∧ (T+, T−, T0) and (ρiT+, ρiT−, ρiT0) respectively (i.e. L+ = (T1T+) , L− = ∧ ′ (T1T−) and so on). By induction, L0 ∼ L0. Therefore, by the definition ′ ′ of skein equivalence, L+ ∼ L+ if and only if L− ∼ L−. In other words, switching a crossing of T2 does not affect the truth or falsity of the assertion. Since T2 can be untied by switching crossings, we are back in the case n ≤ 1. 

4 Mutant links with different Khovanov homology

Let V (L)(q) := Kh(L)(−1,q) denote the graded Euler characteristic of C(D) and W (L)(t) := Kh(L)(1,q) the ordinary (ungraded) Poincar´epolynomial. As is shown in [Kh], V is just an unnormalized version of the Jones polyno- mial. By the results of sections 2 and 3, the Jones polynomial is invariant under Conway mutation. On the other hand, the following theorem gives an example of mutant links which are separated by W .

Theorem 3 Let Ki (i = 1, 2) be a (2,ni) torus link, ni > 2. Then the oriented links

′ L := ⊔ (K1♯K2) and L := K1 ⊔ K2 are Conway mutants with different W polynomial. Here, denotes the trivial knot and K1♯K2 is the connected sum of the oriented links K1 and K2. Note that the connected sum is well-defined even if Ki has two components, because in this case the link Ki is symmetric in its components.

Proof. From Figure 3 it is apparent that L and L′ are Conway mutants. The Khovanov complex of the trivial knot is

. . . −→ 0 −→ 0 −→ A −→ 0 −→ 0 −→ . . .

3 L L’

KK1 2 K1 K2

TT1 2 TT1 h1 2

Figure 3: L and L′ are Conway mutants and rank(A) = 2, whence W ( ) = 2. By [Kh, Proposition 33], Kh is multiplicative under disjoint union, and so W (L) = 2W (K1♯K2). On the other hand, by [Kh, Proposition 35],

−2 −3 −(ni−1) −ni W (Ki)=2+ t + t + . . . + t + t if ni is odd and

−2 −3 −(ni−1) −ni W (Ki)=2+ t + t + . . . + t + 2t

′ if ni is even. Since ni > 2, W (Ki) is not divisible by 2. But then W (L )= ′ W (K1)W (K2) is not divisible by 2 and hence W (L ) 6= W (L). 

Theorems 1, 2 and 3 immediatly imply: Corollary 1 The W polynomial does not satisfy a relation of the kind men- tioned in section 2.

Remark. Theorem 3 remains true if we also allow torus links Ki with ni < −2 (this may be seen using [Kh, Corollary 11], which relates the Khovanov homology of a link to the Khovanov homology of its mirror image). The condition |ni| > 2 is necessary. In fact, if one of the |ni| is ≤ 1, then the ′ corresponding torus link Ki is trivial and hence L and L are isotopic. If ′ n2 = 2, then L and L look as is shown in Figure 4. Note that both L − L0 ′ ′ and L − L0 are isotopic to the link ⊔ K1. Using [Kh, Corollary 10], one can show that both Hi,j(L) and Hi,j(L′) are isomorphic to Hi+2,j+5( ⊔ i,j+1 K1) ⊕H ( ⊔ K1). The cases n2 = −2 and n1 = ±2 are similar.

L L’

K K 1 L 0 1 ’ L 0

′ Figure 4: L and L for the case n2 = 2

4 5 Computer Calculations with KHOHO

Tables 1 and 2 show the Khovanov homology of L and L′ for the case n1 = n2 = 3. The tables where generated using A. Shumakovitch’s program a[b] KhoHo [Sh]. The entry in the i-th column and the j-th row looks like , c where a is the rank of the homology group Hi,j, b the number of factors Z/2Z in the decomposition of Hi,j into p-subgroups, and c the rank of the chain group Ci,j. The numbers above the horizontal arrows denote the ranks of the chain differentials. In the examples, only 2-torsion occurs. It has been conjectured by A. Shu- makovitch that this is actually true for arbitrary links. The reader may verify that not only the dimensions but also the torsion parts of the Hi,j are different for L and L′. The dimensions of the Ci,j agree because there is a natural one-to-one cor- respondence between the Kauffman states of L and L′ (which re-proves the fact that the Jones polynomial is invariant under Conway mutation). We do not know the answer to the following question: Question: Does there exist a pair of mutant oriented knots with distinct Khovanov homology? According to the database of A. Shumakovitch, no such pair of knots with 13 or less crossings exists. In particular, the Kinoshita-Terasaka knot and the Conway knot (the knots depicted in Figure 5) are mutant knots with the same Khovanov homology (see Table 3).

Figure 5: The Kinoshita-Terasaka knot and the Conway knot

References

[Ba] Dror Bar-Natan, On Khovanov’s Categorification of the Jones poly- nomial, Algebraic and Geometric Topology 2-16, 337-370, 2002, arXiv:math.QA/0201043.

[Co] John H. Conway, An enumeration of knots and links and some of their related properties, Computational problems in Abstract Algebra, Pergamon Press, New York, 329-358, 1970.

5 [Ja] Magnus Jacobsson, An invariant of link cobordisms from Khovanov’s homology theory, arXiv:math.GT/0206303v1.

[Ka] Akio Kawauchi, A Survey of Knot Theory, Birkh¨auser, 1996.

[Kh] Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3), 359-426, 1999, arXiv:math.QA/9908171.

[Mu] J. Murakami, The Parallel Version of Polynomial Invariants of Links, Osaka J. Math. 26, 1-55, 1989.

[Sh] Alexander Shumakovitch, KhoHo, Basel 2002, see http://www. geometrie.ch/KhoHo.

[Vi] Oleg Viro, Remarks on the definition of Khovanov Homology, arXiv:math.GT/0202199.

6 -6 -5 -4 -3 -2 -1 0

-2 1 1 -4 0 −−→20 −−→ 40 −−→ 2 2 2 6 6 4 -6 0 −−→10 −−→ 50 −−→ 100 −−→ 182 −−→ 130 −−→ 5 1 1 6 15 28 33 18 6 6 24 36 382[2] 14 4 -8 0 −−→ 0 −−→ 0 −−→ 0 −−→ −−→ 0 −−→ 0 6 30 60 74 54 18 4 15 45 44 280[2] 5 1 -10 0 −−→ 0 −−→ 1 −−→ 2 −−→ −−→ 0 −−→ 0 15 60 90 74 33 6 1 20 391[1] 20 6 -12 0 −−→ 1 −−→ −−→ 2 −−→ 0 20 60 60 28 6 152[1] 130[1] 2 -14 0 −−→ −−→ −−→ 0 15 30 15 2 5 1[1] -16 1 −−→ 6 6 -18 1 1

Table 1: Ranks of Hi,j and Ci,j and ranks of the chain differentials for the disjoint union of the unknot and the granny-knot

7 -6 -5 -4 -3 -2 -1 0

-2 1 1 -4 0 −−→20 −−→ 40 −−→ 2 2 2 6 6 4 -6 0 −−→10 −−→ 50 −−→ 100 −−→ 182 −−→ 130 −−→ 5 1 1 6 15 28 33 18 6 6 24 36 382[2] 14 4 -8 0 −−→ 0 −−→ 0 −−→ 0 −−→ −−→ 0 −−→ 0 6 30 60 74 54 18 4 15 45 44 280[2] 5 1 -10 0 −−→ 0 −−→ 1 −−→ 2 −−→ −−→ 0 −−→ 0 15 60 90 74 33 6 1 20 400[2] 20 6 -12 0 −−→ 0 −−→ −−→ 2 −−→ 0 20 60 60 28 6 152[1] 130[1] 2 -14 0 −−→ −−→ −−→ 0 15 30 15 2 6 0[2] -16 0 −−→ 6 6 -18 1 1

Table 2: Ranks of Hi,j and Ci,j and ranks of the chain differentials for the disjoint union of two trefoil knots

8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 10 50 100 100 6 1 0 9 −−−→ −−−→ −−−→ −−→ −−→ 1 −−→ 1 6 15 20 16 8 1 0 10 120 550 1250 1540 107 41 4 0 7 −−−→ −−−−→ −−−→ −−−→ −−−→ −−→ 1 −−→ 0[1] −−→ 1 13 67 180 279 261 149 45 4 0 50 550 2600 6580 9510 792 371 890 6 0 5 −−−→ −−−→ −−−−→ −−−→ −−−→ −−−→ 1 −−→ 1[1] −−→ −−→ 5 60 315 918 1609 1743 1164 461 95 6 0 100 1260 6630 18950 3159 3104 1767 5650 910 4 0 3 −−−→ −−−→ −−−→ −−−−→ −−−→ 1 −−−→ 2 −−−→ 1[1] −−→ −−→ −−→ 10 136 789 2558 5054 6264 4873 2333 656 95 4 0 100 1580 9830 32410 6238 7166 4871 19160 4170 440 1 0 1 −−→ −−−→ −−−→ −−−→ −−−−→ 2 −−−→ 1[1] −−−→ 1[2] −−−→ −−→ −−→ −−→ 10 168 1141 4224 9479 13406 12038 6788 2333 461 45 1 0 60 1130 8590 33750 7607 10219 8186 38500 10230 1410 8 0 -1 −−→ −−→ −−−→ −−−→ −−−→ 1 −−−−→ 3[1] −−−→ 2[1] −−−→ −−−→ −−→ −−→ 6 119 972 4234 10982 17827 18408 12038 4873 1164 149 8 0 10 420 4320 21280 5788 9186 8639 47660 14980 2450 16 0 9 -3 −−→ −−→ −−→ −−−→ −−−→ 2 −−−→ 2[1] −−−−→ 1[1] −−−→ −−−→ −−−→ −−→ 1 43 474 2560 7916 14976 17827 13406 6264 1743 261 16 0 50 1130 7840 2708 5207 5774 37040 13500 2590 20 0 -5 −−→ −−→ −−→ −−−→ 1 −−−→ 1[2] −−−→ 1[1] −−−−→ −−−→ −−−→ −−−→ 5 118 897 3492 7916 10982 9479 5054 1609 279 20 0 100 1520 745 1814 2418 18050 7530 1650 15 0 -7 −−→ −−→ −−→ 1 −−−→ 2[1] −−−→ 1 −−−→ −−−−→ −−−→ −−−→ 10 162 897 2560 4234 4224 2558 918 180 15 0 100 108 365 6060 5350 2540 610 6 0 -9 −−→ −−→ 1 −−→ 1[1] −−−→ −−−→ −−−→ −−−−→ −−−→ 10 118 474 972 1141 789 315 67 6 0 50 38 800 880 480 120 1 0 -11 −−→ −−→ 1[1] −−→ −−−→ −−−→ −−−→ −−−−→ 5 43 119 168 136 60 13 1 0 1460 0 40 1 0 -13 −−→ 1 −−→ −−→ −−−→ −−−→ 1 6 10 10 5 1

Table 3: Ranks of Hi,j and Ci,j and ranks of the chain differentials for either the Kinoshita-Terasaka knot or the Conway knot (both have the same Khovanov homology)