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 invari- ants 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 fi- nite 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 defi- nition 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 ´is 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. ST U 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, ST U 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.