FINITE TYPE INVARIANTS 1. Introduction Knots Belong to Sailors and Climbers and Upon Further Reflection, Perhaps Also to Geomete

Home , Finite type invariant, HOMFLY polynomial, Jones polynomial, Knot theory, Kontsevich invariant, Quantum invariant

FINITE TYPE INVARIANTS

DROR BAR-NATAN

Abstract. This is an overview article on ﬁnite type invariants, written for the Encyclopedia of Mathematical Physics.

1. Introduction structure and are deeply related to Lie algebras (Sec- tion 2). There are several constructions for a “univer- Knots belong to sailors and climbers and upon sal finite type invariant” and those are related to con- further reflection, perhaps also to geometers, topolo- formal field theory, the Chern-Simons-Witten topo- gists or combinatorialists. Surprisingly, throughout logical quantum field theory and Drinfel’d’s theory the 1980s it became apparent that knots are also of associators and quasi-Hopf algebras (Section 3). closely related to several other branches of mathe- Finite type invariants have been studied extensively matics in general and mathematical physics in par- (Section 4) and generalized in several directions (Sec- ticular. Many of these connections (though not tion 5). But the first question on finite type invariants all!) factor through the notion of “finite type in- remains unanswered: variants” (aka “Vassiliev” or “Goussarov-Vassiliev” invariants) [Go1, Go2, Va1, Va2, BL, Ko1, BN2]. Problem 2. Honest polynomials are dense in the Let V be an arbitrary invariant of oriented knots space of functions. Are finite type invariants dense in oriented space with values in some Abelian group within the space of all knot invariants? Do they sep- A. Extend V to be an invariant of 1-singular knots, arate knots? knots that may have a single singularity that locally looks like a double point , using the formula In a similar way one may define finite type invariants of framed knots (and ask the same questions). (1) V ( ) = V (!) − V ("). 1.1. Acknowledgement. I wish to thank O. Das- m Further extend V to the set K of m-singular knots bach and A. Savage for comments and corrections. (knots with m double points) by repeatedly using (1).

Definition 1. We say that V is of type m if its ex- 2. Basic facts tension V |Km+1 to (m + 1)-singular knots vanishes identically. We say that V is of finite type if it is of 2.1. Classical knot polynomials. The first (non type m for some m. trivial!) thing to notice is that there are plenty of finite type invariants and they are at least as powerful Repeated differences are similar to repeated as all the standard knot polynomials1 combined: derivatives and hence it is fair to think of the definition of V |Km as repeated differentiation. With this Theorem 3. [BN2, BL] Let J(K)(q) be the Jones in mind, the above definition imitates the definition polynomial of a knot K (it is a Laurent polynomial of polynomials of degree m. Hence finite type invari- in a variable q). Consider the power series expansion x P∞ m ants can be thought of as “polynomials” on the space J(K)(e ) = m=0 Vm(K)x . Then each coefficient of knots. Vm(K) is a finite type knot invariant. (And thus the As we shall see below, finite type invariants are Jones polynomial can be reconstructed from finite type plenty and powerful and they carry a rich algebraic information).

Date: First edition: Aug. 13, 2004. This edition: Nov. 10, 2008 . This work was partially supported by NSERC grant RGPIN 262178. Electronic versions: http://www.math.toronto.edu/ ∼drorbn/papers/EMP/ and arXiv:math.GT/0408182. Copyright is retained by the author. 1Finite type invariants are like polynomials on the space of knots; the standard phrase “knot polynomials” refers to a diﬀerent thing — knot invariants with polynomial values. 1 2 DROR BAR-NATAN

A similar theorem holds for the Alexander- 2.3. The structure of A. Knots can be multiplied Conway, HOMFLY-PT and Kauffman polynomi- (the “connected sum” operation) and knot invariants als [BN2], and indeed, for arbitrary Reshetikhin- can be multiplied. This structure interacts well with Turaev invariants [RT], [Lin1]. Though it is still un- finite type invariants and induces the following struc- known if the signature of a knot can be expressed in ture on Ar and A: terms of its finite type invariants. Theorem 6. [Ko1, BN2, Wil, CDL] Ar and A 2.2. Chord diagrams and the Fundamental are commutative and cocommutative graded bialge- Theorem. The top derivatives of a multi-variable bras (i.e., each carries a commutative product and polynomial form a system of constants that determine a compatible cocommutative coproduct). Thus both r that polynomial up to polynomials of lower³ degree.´ A and A are graded polynomial algebras over their m r r Likewise the mth derivative V (m) := V ··· spaces of primitives, P = ⊕mPm and P = ⊕mPm. of³ a type m´ invariant³ V ís a³ constant´ (for m m m+1 Framed knots differ from knots only by a single V ··· ! −V ··· " = V ··· = integer parameter (the “self linking”, itself a type 1 0 so V (m) is blind to 3D topology) and likewise V (m) invariant). Thus Pr and P are also closely related. determines V up to invariants of lower type. Hence r a primary tool in the study of finite type invariants Theorem 7. [BN2] P = P ⊕ hθi, where θ is the is the study of the “top derivative” V (m), also known unique 1-chord diagram . as “the weight system of V ”. 2.4. Bounds and computational results. The Blind to 3D topol- 3 following table (taken from [BN2, Kn]) shows the ogy, V (m) only sees the 4 3 number of type m invariants of knots and framed combinatorics of the 2 2 1 knots modulo type m − 1 invariants (dim Ar and circle that parametrizes 1 m 4 dim A ) and the number of multiplicative generators an m-singular knot. On m of the algebra A in degree m (dim P ) for m ≤ 12. this circle there are m pairs of points that are pair- m Some further tabulated results are in [BN5]. wise identified in the image; standardly one indicates those by drawing a circle with m chords marked (an m 0 1 2 3 4 5 6 7 8 9 10 11 12 r “m-chord diagram”) as on the right. dim Am 1 0 1 1 3 4 9 14 27 44 80 132 232 dim Am 1 1 2 3 6 10 19 33 60 104 184 316 548 Definition 4. Let Dm denote the space of all for- dim Pm 0 1 1 1 2 3 5 8 12 18 27 39 55 mal linear combinations with rational coefficients of r Little is known about these dimensions for large m-chord diagrams. Let Am be the quotient of Dm by all 4T and FI relations as drawn below (full details m. There is an explicit conjecture in [Br] but no in e.g. [BN2]), and let Aˆr be the graded completion of progress has been made in the direction of proving or L disproving it. The best asymptotic bounds available A := Ar . Let A , A and Aˆ be the same as Ar , m m m m are: Ar and Aˆr but without imposing the FI relations. √ c m Theorem 8. Forq large m, dim Pm > e (for 4T : FI : 2 any fixed c < π 3 ) [Da, Ko3] and dim Am < m √ 2m Theorem 5. (The Fundamental Theorem) 6 m! m/π [Sto, Za]. • (Easy part, [Va1, Go1, BL]). If V is a rational valued type m invariant then V (m) defines a linear 2.5. Jacobi diagrams and the relation with Lie r (m) algebras. Much of the richness of finite type invari- functional on Am. If in addition V ≡ 0, then V is of type m − 1. ants stems from their relationship with Lie algebras. • (Hard part, [Ko1] and Section 3). For any linear Theorem 9 below suggests this relationship on an ab- r stract level, Theorem 10 makes that relationship con- functional W on Am there is a rational valued type m invariant V so that V (m) = W . crete and Theorem 12 makes it a bit deeper. Thus to a large extent the study of finite type in- Theorem 9. [BN2] The algebra A is isomorphic to variants is reduced to the finite (though super ex- the algebra At generated by “Jacobi diagrams in a r ponential in m) algebraic study of Am. A similar circle” (chord diagrams that are also allowed to have theorem reduces the study of finite type invariants of oriented internal trivalent vertices) modulo the AS, framed knots to the study of Am. STU and IHX relations. See the figure below. FINITE TYPE INVARIANTS 3

Note that B can be graded (by half the number of vertices in a Jacobi diagram) and that χ respects AS: + =0 degrees so it extends to an isomorphism χ : Bˆ → Aˆ STU: = − of graded completions.

IHX: = − 3. Proofs of the Fundamental Theorem A Jacobi diagram in a circle The heart of all known proofs of Theorem 5 is al- Thinking of trivalent vertices as graphical analogs ways a construction of a “universal finite type invari- of the Lie bracket, the AS relation become the anti- ant” (see below); it is simple to show that the exis- commutativity of the bracket, STU become the equa- tence of a universal finite type invariant is equivalent tion [x, y] = xy − yx and IHX becomes the Jacobi to Theorem 5. identity. This analogy is made concrete within the proof of the following: Definition 13. A universal finite type invariant is a map Z : {knots} → Aˆr whose extension to singular Theorem 10. [BN2] Given a finite dimensional knots satisfies Z(K) = D + (higher degrees) when- metrized Lie algebra g (e.g., any semi-simple Lie al- ever a singular knot K and a chord diagram D are gebra) there is a map T : A → U(g)g defined on A g related as in Section 2.2. and taking values in the invariant part U(g)g of the universal enveloping algebra U(g) of g. Given also 3.1. The Kontsevich Integral. The first construc- a finite dimensional representation R of g there is a tion of a universal finite type invariant was given by linear functional Wg,R : A → Q. Kontsevich [Ko1] (see also [BN2, CD]). It is known as The last assertion along with Theorem 5 show that “the Kontsevich Integral” and up to a normalization associated with any g, R and m there is a weight factor it is given by2 system and hence a knot invariant. Thus knots are X∞ XZ ^m 0 1 dzi − dz unexpectedly linked with Lie algebras. Z (K) = (−1)#P↓ D i , 1 (2πi)m P z − z0 The hope [BN2] that all finite type invariants arise m=0 i=1 i i t1<...

3.2. Perturbative Chern-Simons-Witten the- Problem 14. Does the perturbative expansion of the ory and configuration space integrals. Histor- Chern-Simons-Witten theory converge (or is asymp- ically the first approach to the construction of a totic to) the exact solution due to Witten [Wit] and universal finite type invariant was to use perturba- Reshetikhin-Turaev [RT] when the parameter k con- tion theory with the Chern-Simons-Witten topologi- verges to infinity? cal quantum field theory; this is also how the relationship with Lie algebras first arose [BN1]. But taming 3.3. Associators and trivalent graphs. There is the integrals involved turned out to be difficult and also an entirely algebraic approach for the construc- working constructions using this approach appeared tion of a universal finite type invariant Z3. The idea is only a bit later [BoTa, Th1, AF]. to find some algebraic context within which knot the- In short, one writes a perturbative expansion for ory is finitely presented — i.e., presented by finitely the large k asymptotics of the Chern-Simons-Witten many generators subject to finitely many relations. If path integral for some metrized Lie algebra g with a the algebraic context at hand is compatible with the Wilson loop in some representation R of g, definitions of finite type invariants and of chord dia- Z · ¸ grams, one may hope to define Z3 by defining it on the R generators in such a way that the relations are satis- ik 2 DA trRholK (A) exp 4π tr(A∧dA+ 3 A∧A∧A) . g-connections R3 fied. Thus the problem of defining Z3 is reduced to finding finitely many elements of A-like spaces which The result is of the form solve certain finitely many equations. Z X X A concrete realization of this idea is in [LM2, BN4] Wg(D),R E(D) (following ideas from [Dr1, Dr2] on quasitriangular D: Feynman diagram Quasi-Hopf algebras). The relevant “algebraic context” is a category with certain extra operations, and where E(D) is a very messy integral expression and within it, knot theory is generated by just two el- the diagrams D as well as the weights Wg(D),R are ements, the braiding ! and the re-association 8. as in Section 2.5. Replacing W by simply D in g(D),R Thus to define Z3 it is enough to find R = Z3(!) the above formula we get an expression with values and “an associator” Φ = Z3(8) which satisfy certain in Aˆ: X XZ normalization conditions as well as the pentagon and Z2(K) := D E(D) ∈ Aˆ. hexagon equations D Φ123 · (1∆1)(Φ) · Φ234 = (∆11)(Φ) · (11∆)(Φ),

For formal reasons Z2(K) ought to be a univer- ± 123 ± 23 −1 132 ± 13 312 sal finite type invariant, and after much work taming (∆1)(R ) = Φ (R ) (Φ ) (R ) Φ . the E(D) factors and after multiplying by a further As it turns out the solution for R is easy and nearly framing-dependent renormalization term Zanomaly, canonical. But finding an associator Φ is a lot harder. the result is indeed a universal finite type invariant. There is a closed form integral expression ΦKZ due Upon further inspection the E(D) factors can be to [Dr1] but one encounters the same not-too-well- reinterpreted as integrals of certain spherical vol- understood multiple ζ numbers of Section 3.1. There ume forms on certain (compactified) configuration is a rather complicated iterative procedure for finding spaces [BoTa]. These integrals can be further inter- an associator [Dr2, BN8], [BN4]. On a computer it preted as counting certain “tinker toy constructions” had been used to find an associator up to degree 7. built on top of K [Th1]. The latter viewpoint makes There is also closed form associator that works only the construction of Z2 visually appealing [BN9], but with the Lie super-algebra gl(1|1) [Lie2]. But it re- there is no satisfactory writeup of this perspective mains an open problem to find a closed form formula yet. for a rational associator (existence by [Dr2, BN8]). We note that the precise form of the renormaliza- On the positive side we should note that the end anomaly tion term Z remains an open problem. An result, the invariant Z3, is independent of the choice anomaly 1 % appealing conjecture is that Z = exp 2 . If of Φ and that Z3 = Z1 [LM2]. this is true then Z2 = Z1 [Po]; but the conjecture There is an alternative (more symmetric and in- is only verified up to degree 6 [Les] (there’s also an trinsicly 3-dimensional, but less well documented) unconfirmed verification to all orders [Ya]). description of the theory of associators in terms of The most important open problem about pertur- knotted trivalent graphs [BNT], [Th2]. There ought bative Chern-Simons-Witten theory is not directly to be a perturbative invariant associated with knot- about finite type invariants, but it is nevertheless ted trivalent graphs in the spirit of Section 3.2 and worthwhile to recall it here: such an invariant should lead to a simple proof that FINITE TYPE INVARIANTS 5

Z2 = Z3 = Z1. But the E(D) factors remain untamed in it correspond to finite type invariants. This later in this case. got related to the Goodwillie calculus and back to the configuration spaces of Section 3.2. See [Vol]. 3.4. Step by step integration. The last approach for proving the Fundamental Theorem is the most natural and historically the first. But here it is last 4.2. Interdependent modifications. The stan- because it is yet to lead to an actual proof. A weight dard definition of finite type invariants is based on r system W : Am → Q is an invariant of m-singular modifying a knot by replacing over (or under) cross- knots. We want to show that it is the mth derivative ings with under (or over) crossings. In [Go3] Gous- of an invariant V of non-singular knots. It is natu- sarov generalized this by allowing arbitrary modifi- ral to try to integrate W step by step, first finding an cations done to a knot — just take any segment of invariant V m−1 of (m−1)-singular knots whose deriv- the knot and move it anywhere else in space. The ative in the sense of (1) is W , then an invariant V m−2 resulting new “finite type” theory turns out to be of (m − 2)-singular knots whose derivative is V m−1, equivalent to the old one though with a factor of 2 and so on all the way up to an invariant V 0 = V applied to the grading (so an “old” type m invariant whose mth derivative will then be W . If proven, the is a “new” type 2m invariant and vice versa). See following conjecture would imply that such an induc- also [BN10, Co]. tive procedure can be made to work: Conjecture 15. [Hu] If V r is a once-integrable in- 4.3. n-equivalence, commutators and claspers. variant of r-singular knots then it is also twice in- While little is known about the overall power of finite tegrable. That is, if there is an invariant V r−1 of type invariants, much is known about the power of (r − 1)-singular knots whose derivative is V r, then type n invariants for any given n. Goussarov [Go2] there is an invariant V r−2 of (r − 2)-singular knots defined the notion of n-equivalence: two knots are whose second derivative is V r. said to be “n-equivalent” if all their type n invariants are the same. This equivalence relation is well In [Hu] Hutchings reduced this conjecture to a understood both in terms of commutator subgroups certain appealing topological statement and further of the pure braid group [Sta, NS] and in terms of to a certain combinatorial-algebraic statement about Habiro’s calculus of surgery over “claspers” [Ha] (the 1 the vanishing of a certain homology group H which latter calculus also gives a topological explanation for is probably related to Kontsevich’s graph homol- the appearance of Jacobi diagrams as in Section 2.5). 0 ogy complex [Ko2] (Kontsevich’s H is A, so this In particular, already Goussarov [Go2] shows that is all in the spirit of many deformation theory prob- the set of equivalence classes of knots modulo n- 0 lems where H enumerates infinitesimal deformations equivalence is a finitely generated Abelian group G 1 n and H is the obstruction to globalization). Hutch- under the operation of connected sum, and the rank 1 ings [Hu] was also able to prove the vanishing of H of that group is equal to the dimension of the space (and hence reprove the Fundamental Theorem) in the of type n invariants. simpler case of braids. But no further progress has Ng [Ng] has shown that ribbon knots generate an been made along these lines since [Hu]. index 2 subgroup of Gn. 4. Some further directions 4.4. Polynomiality and Gauss sums. Gous- We would like to touch on a number of significant sarov [Go4] (see also [GPV]) found an intriguing way further directions in the theory of finite type invari- to compute finite type invariants from a Gauss di- ants. We will only say a few words on each of those agram presentation of a knot, showing in particular and refer the reader to the literature for further in- that finite type invariants grow as polynomials in the formation. number of crossings n and can be computed in poly- 4.1. The original “Vassiliev” perspective. nomial time in n (though actual computer programs V.A. Vassiliev came to the study of finite type knot are still missing!). invariants by studying the infinite dimensional space Gauss diagrams are obtained from knot diagrams of all immersions of a circle into R3 and the topol- in much of the same way as Chord diagrams are ob- ogy of the “discriminant”, the locus of all singular tained from singular knots, except all crossings are immersions within the latter space [Va1, Va2]. Vas- counted and not just the double points, and certain siliev studied the topology of the complement of the over/under and sign information is associated with discriminant (the space of embeddings) using a cer- each crossing/chord so that the knot diagram can be tain spectral sequence and found that certain terms recovered from its Gauss diagram. In the example 6 DROR BAR-NATAN

below, we also dashed a subdiagram of the Gauss di- 4.6. Taming the Kontsevich Integral. While ex- agram equivalent to the chord diagram : plicit calculations are rare, there is a nice structure theorem for the values of the Kontsevich integral, say- ing that for a knot K and up to any fixed number 4 3 −1 4 of loops in the Jacobi diagrams, χ Z1(K) can be 5 − 2 − 3 described by finitely many rational functions (with − + + 1 1 + denominators powers of the Alexander polynomial) 5 6 which dictate the placement of the legs. This struc- + + 2 2 5 + − ture theorem was conjectured in [Ro], proven in [Kr] + 1 6 6 and partially generalized to links in [GK]. + 4 3 4.7. The Rozansky-Witten theory. One way to If G is a Gauss diagram and D is a chord diagram construct linear functionals on A (and hence finite we let hD,Gi be the number of subdiagrams of G type invariants) is using Lie algebras and represen- equivalent to D, counted with appropriate signs (to tations as in Section 2.5; much of our insight about be precise, we also need to base the diagrams involved A comes this way. But there is another construc- and count subdiagrams that respect the basing). tion for such functionals (and hence invariants), due Theorem 16. [Go4, GPV] If V is a type m invariant to Rozansky and Witten [RozWit], using contrac- then there are finitely many (based) chord diagrams tions of curvature tensors on hyper-Kählermanifolds. Very little is known about the Rozansky-Witten ap- Di with at mostPm chords and rational numbers αi so that V (K) = i αihDi,Gi whenever G is a Gauss proach; in particular, it is not known if it is stronger diagram representing a knot K. or weaker than the Lie algebraic approach. For an application of the Rozansky-Witten theory back to 4.5. Computing the Kontsevich Integral. While hyper-Kählergeometry check [HS] and for a unifi- the Kontsevich Integral Z1 is a cornerstone of the the- cation of the Rozansky-Witten approach with the ory of finite type invariants, it has been computed for Lie algebraic approach (albeit at a categorical level) surprisingly few knots. Even for the unknot the result check [RobWil]. is non-trivial: Theorem 17. (“Wheels”, [BGRT1, BLT]) The 4.8. The Melvin-Morton conjecture and the F framed Kontsevich integral of the unknot, Z1 (°), ex- volume conjecture. The Melvin-Morton conjec- pressed in terms of diagrams in Bˆ, is given by Ω = ture (stated [MeMo], proven [BNG]) says that P∞ the Alexander polynomial can be read off cer- exp∪· n=1 b2nω2n, where the ‘modified Bernoulli numbers’ b are defined by the power series expan- tain coefficients of the coloured Jones polynomial. P 2n ∞ 2n 1 sinh x/2 The Kashaev-Murakami-Murakami volume conjec- sion n=0 b2nx = 2 log x/2 , the ‘2n-wheel’ ω2n is the free Jacobi diagram made of a 2n-gon with 2n ture (stated [Ka, MuMu], unproven) says that a certain asymptotic growth rate of the coloured Jones legs (so e.g., ω = ) and where exp means ‘ex- 6 ∪· polynomial is the hyperbolic volume of the knot com- ponential in the disjoint union sense’. plement. Closed form formulas have also been given for the Both conjectures are not directly about finite type Kontsevich integral of framed unknots, the Hopf link invariants but both have ramifications to the theory and Hopf chains [BNL] and for torus knots [Ma]. of finite type invariants. The Melvin-Morton conjec- Theorem 17 has a companion that utilizes the ture was first proven using finite type invariants and same element Ω, the “wheeling” theorem [BGRT1, several later proofs and generalizations (see [BN6]) BLT]. The wheeling theorem “upgrades” the vector also involve finite type invariants. The volume conjec- space isomorphism χ : B → A to an algebra isomor- ture would imply, in particular, that the hyperbolic phism and is related to the Duflo isomorphism of the volume of a knot complement can be read from that theory of Lie algebras. It is amusing to note that knot’s finite type invariants, and hence finite type in- the wheeling theorem (and hence Duflo’s theorem in variants would be at least as strong as the volume the metrized case) follows using finite type techniques invariant. from the “1 + 1 = 2 on an abacus” identity A particularly noteworthy result and direction for further research is Gukov’s [Gu] recent unification of these two conjectures under the Chern-Simons um- # . brella (along with some relations to three dimensional quantum gravity). FINITE TYPE INVARIANTS 7

5. Beyond knots [BN5] D. Bar-Natan, Some computations related to Vas- siliev invariants, electronic publication, http: For the lack of space we have restricted ourselves //www.math.toronto.edu/∼drorbn/LOP.html# here to a discussion of finite type invariants of knots. Computations. But the basic “differentiation” idea of Section 1 calls [BN6] D. Bar-Natan, Bibliography of Vassiliev invariants, for generalization, and indeed it has been generalized http://www.math.toronto.edu/∼drorbn/VasBib. [BN7] D. Bar-Natan, Vassiliev and quantum invariants of extensively. We will only make a few quick comments. braids, in Proc. of Symp. in Appl. Math. 51 (1996) 129– Finite type invariants of homotopy links (links 144, The interface of knots and physics, (L. H. Kauff- where each component is allowed to move across it- man, ed.), Amer. Math. Soc., Providence. self freely) and of braids are extremely well behaved. [BN8] D. 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Thurston, The Arhus˚ integral of rational ho- [LMO, Le] and then [BGRT2, KT]), they all agree or mology 3-spheres I–III, Selecta Mathematica, New Se- are conjectured to agree, and they are related to the ries 8 (2002) 315–339, arXiv:q-alg/9706004, 8 (2002) Chern-Simons-Witten theory. 341–371, arXiv:math.QA/9801049, 10 (2004) 305–324, Finite type invariants were studied for several arXiv:math.QA/9808013. other types of topological objects, including knots [BNL] D. Bar-Natan and R. Lawrence, A Rational Surgery Formula for the LMO Invariant, Israel Jour- within other manifolds, higher dimensional knots, nal of Mathematics 140 (2004) 29–60, arXiv: virtual knots, plane curves and doodles and more. math.GT/0007045. See [BN6]. [BLT] D. Bar-Natan, T. Q. T. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geometry and Topology 7-1 (2003) 1–31, arXiv:math.QA/0204311. References [BNT] D. Bar-Natan and D. 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Department of Mathematics, University of Toronto, Toronto Ontario M5S 3G3, Canada E-mail address: [email protected] URL: http://www.math.toronto.edu/~drorbn