Integral Expressions for the Vassiliev Knot Invariants
Dylan P. Thurston
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51 pages, 25 figures. AB thesis, Harvard University, Spring 1995. Note that there are problems, notably references that should be added and an incorrect treatment of the anomaly for non-primitive weight systems. Nevertheless, it may be of interest
It has been folklore for several years in the knot theory community that certain integrals on configuration space, originally motivated by perturbation theory for the Chern-Simons field theory, converge and yield knot invariants. This was proposed independently by Gaudagnini, Martellini, and Mintchev and Bar-Natan. The analytic difficulties involved in proving convergence and invariance were reportedly worked out by Bar-Natan, Kontsevich, and Axelrod and Singer.
But I know of no elementary exposition of this fact. ... This thesis is an attempt to remedy this lack. I adopt an almost exclusively topological point of view, rarely mentioning Chern-Simons theory. There are also a few new results in this thesis. These include a new construction of the functorial compactification of configuration space (Section 3.2) as well as some variations on the integrals. For a suitable choice of this variation, the integral reduces to counting tinkertoy diagrams (Section 4.5). In particular, the invariants constructed take values in Q.
Vassiliev's knot invariants can be computed in different ways but many of them as Kontsevich integral are very difficult. We consider more visual diagram formulas of the type Polyak-Viro and give new diagram formula for the two basic Vassiliev invariant of degree 4.
We show that the number of homomorphisms from a knot group to a finite group $G$ cannot be a Vassiliev invariant, unless it is constant on the set of $(2,2p+1)$ torus knots. In several cases, such as when $G$ is a dihedral or symmetric group, this implies that the number of homomorphisms is not a Vassiliev invariant.
We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about knots modulo the nth derived subgroups of the pure braid groups, and about knots modulo braid subgroups in general.
We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.
In this paper we prove that the $r$-th ADO polynomial of a knot, for $r$ a power of prime number, can be expanded as Vassiliev invariants with values in $\mathbb{Z}$. Nevertheless this expansion is not unique and not easily computable. We can obtain a unique computable expansion, but we only get $r$ adic topological Vassiliev invariants as coefficients. To do so, we exploit the fact that the colored Jones polynomials can be decomposed as Vassiliev invariants and we tranpose it to ADO using the unified knot invariant recovering both ADO and colored Jones defined in arXiv:2003.09854. Finally we …
Vassiliev invariants up to order six for arbitrary torus knots $\{ n , m \}$, with $n$ and $m$ coprime integers, are computed. These invariants are polynomials in $n$ and $m$ whose degree coincide with their order. Furthermore, they turn out to be integer-valued in a normalization previously proposed by the authors.
We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on $g+1$ parameters $n_1,\ldots,n_{g+1}$. These invariants are symmetric polynomials in $n_1,\ldots,n_{g+1}$ whose degree coincide with their order. We also discuss their topological and integer-valued properties.
A `total Chern class' invariant of knots is defined. This is a universal Vassiliev invariant which is integral `on the level of Lie algebras' but it is not expressible as an integer sum of diagrams. The construction is motivated by similarities between the Kontsevich integral and the topological Chern character.
At any order, the perturbative expansion of the expectation values of Wilson lines in Chern-Simons theory gives certain integral expressions. We show that they all lead to knot invariants. Moreover these are finite type invariants whose order coincides with the order in the perturbative expansion. Together they combine to give a universal Vassiliev invariant.
The values that the first two Vassiliev invariants take on prime knots with up to fourteen crossings are considered. This leads to interesting fish-like graphs. Several results about the values taken on torus knots are proved.
We prove a refinement of Vogel's statement that the Vassiliev invariants of knots coming from semisimple Lie algebras do not generate all Vassiliev invariants. This refinement takes into account the second grading on Vassiliev invariants induced by cabling of knots. As an application we get an amelioration of the actually known lower bounds for the dimensions of the space of Vassiliev invariants.
We analyse the perturbative series expansion of the vacuum expectation value of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel of a Vassiliev invariant of order n is not a knot invariant, since it depends on the regular knot projection chosen, but it differs from a Vassiliev invariant by terms that vanish on knots with n singular crossings. We conjecture that Vassiliev invariants can be reconstructed from their kernels. We present the general form of the kernel of a Vassiliev invariant an…
We introduce a Poincaré polynomial with two-variable $t$ and $x$ for knots, derived from Khovanov homology, where the specialization $(t, x)$ $=$ $(1, -1)$ is a Vassiliev invariant of order $n$. Since for every $n$, there exist non-trivial knots with the same value of the Vassiliev invariant of order $n$ as that of the unknot, there has been no explicit formulation of a perturbative knot invariant which is a coefficient of $y^n$ by the replacement $q=e^y$ for the quantum parameter $q$ of a quantum knot invariant, and which distinguishes the above knots together with the unknot. The first formu…
We prove that the knot invariant induced by a $\Bbb Z$-homology 3-sphere invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order $\leq k-2$. The method developed in our computation shows that there is no $\Bbb Z$-homology 3-sphere invariant of order 5. This result agrees with a conjecture of Rozansky based on physical predictions about the asymptotic behavior of Witten's Chern-Simons path integral.
Given a positive integer $n$, we say that two knots are $V_n$-equivalent if they have the same Vassiliev invariants of order $\le n$. We showed that the $V_n$-equivalence classes of ribbon knots form a group, the operation being induced by the connected sums of knots. The group is free abelian and its rank is equal to the dimension of the space of primitive Vassiliev invariants of order $\le n$. The $n$-equivalence classes (following Gusarov's notion) of ribbon knots also constitute a group. As a corollary, any non-ribbon knot whose Arf invariant istrivial cannot be distinguished from ribbon k…
Vassiliev's knot invariants can be computed in different ways but many of them as Kontsevich integral are very difficult. We consider more visual diagram formulas of the type Polyak-Viro and give new diagram formula for the two basic Vassiliev invariant of degree 4.
This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and as a guide to some of the more advanced material. Our aim is to lead the reader to understanding by means of pictures and calculations, and for this reason we often prefer to convey the idea of the proof on an instructive example rather than give a complete argument. While we have made an effort to make the text reasonably self-contained, an advanced reader…
It was shown by Goussarov that Vassiliev invariants are polynomials in the gleams for a fixed Turaev shadow. In this paper we show that Vassiliev invariants are almost characterized by this fact. We also prove that the space of knot invariants which are polynomials in the gleams is bigger than the Vassiliev subspace.
Vassiliev's knot invariants can be computed in different ways but many of them as Kontsevich integral are very difficult. We consider more visual diagram formulas of the type Polyak-Viro and give new diagram formula for the two basic Vassiliev invariant of degree 4
We use an example to provide evidence for the statement: the Vassiliev-Kontsevich invariants $k_n$ of a knot (or braid) $k$ can be redefined so that $k = \sum_0^\infty k_n$. This constructs a knot from its Vassiliev-Kontsevich invariants, like a power series expansion. The example is pure braids on two strands $P_2\cong \mathbb{Z}$, which leads to solving $e^τ=q$ for $τ$ a Laurent series in $q$. We set $τ= \sum_1^\infty (-1)^{n+1} (q^n - q^{-n})/n$ and use Parseval's theorem for Fourier series to prove $e^τ=q$. Finally we describe some problems, particularly a Plancherel theorem for braid grou…
We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M_11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish.
We analyse the structure of the perturbative series expansion of Chern-Simons gauge theory in the light-cone gauge. After introducing a regularization prescription that entails the consideration of framed knots, we present the general form of the vacuum expectation value of a Wilson loop. The resulting expression turns out to give the same framing dependence as the one obtained using non-perturbative methods and perturbative methods in covariant gauges. It also contains the Kontsevich integral for Vassiliev invariants of framed knots.
The best known examples of Vassiliev invariants are the coefficients of a Jones-type polynomial expanded after exponential substitution. We show that for a given knot, the first $N$ Vassiliev invariants in this family determine the rest for some integer $N$.
The purpose of this note is to state some definitions that may be useful in the study of knots, manifolds and the like. They apply to anything for which the concept of a regional change can be defined, such as a product of elements in a group.
We use an explicit isomorphism from the representation ring of the quantum group U_q(sl(N)) to the Homfly skein of the annulus, to determine an element of the skein which is the image of the mth Adams operator, ψ_m, on the fundamental representation, c_1. This element is a linear combination of m very simple m-string braids. Using this skein element, we show that the Vassiliev invariant of degree n in the power series expansion of the U_q(sl(N)) quantum invariant of a knot coloured by ψ_m(c_1) is the canonical Vassiliev invariant with weight system W_nψ_m^{(n)} where W_n is the weight system f…
Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [Bar-Natan, Garoufalidis, Rozansky and Thurston, arXiv:q-alg/9703025] and [Deligne, letter to Bar-Natan, January 1996, http://www.ma.huji.ac.il/~drorbn/Deligne/], which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computa…
The most general R-matrix type state sum model for link invariants is constructed. It contains in itself all R-matrix invariants and is a generating function for "universal" Vassiliev link invariants. This expression is more simple than Kontsevich's expression for the same quantity, because it is defined combinatorially and does not contain any integrals, except for an expression for "the universal Drinfeld's associator".
We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group.
This paper is a little more detailed version of math-QA/0010017 "Sur l'homologie des espaces de nœuds non-compacts", where the first term of the Vassiliev spectral sequence (computing the homology of the space of long knots in ${\mathbb R}^d$, $d\ge 3$) was described in terms of the Hochschild homology of the Poisson algebras operad for $d$ odd, and of the Gerstenhaber algebras operad for $d$ even. In particular, the bialgebra of chord diagrams arises as some subspace of this homology. The homology in question is the space of characteristic classes for Hochschild cohomology of Poisson (resp. G…
The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M. Hutchings, a geometrical approach following Kontsevich, an algebraic approach following Drinfel'd's theory of associators, and a physical approach coming from the Chern-Simons quantum field theory. Each of these approaches is unsatisfactory in one way or another, and hence we argue that we still don't really understand the fundamental theorem of Vassiliev invari…
The universal Vassiliev invariant from the perturbative Chern-Simons theory is actually a knot invariant without any correction term. The anomaly considered by Bott and Taubes is proved to be zero.
Braided deformations of (symmetric) monoidal categories are related to Vassiliev theory by a direct generalization of well-known results relating "quantum" knot invariants to Vassiliev invariants. The deformation theory of braidings is subsumed by the deformation theory of monoidal functors, which proves surprisingly rich: the deformation complex of a monoidal functor has the same structure as the deformation complex of an algebra, including a pre-Lie structure, from which it is see that the problem of deforming monoidal functors (including braidings) is purely cohomological in nature.
We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for this invariant.
A rational Ansatz is proposed for the generating function $\sum_{j,k} β_{2j+k,2j}x^j y^k$, where $β_{m,u}$ is the number of primitive chinese character diagrams with $u$ univalent and $2m-u$ trivalent vertices. For $P_m:=\sum_{u\ge2}β_{m,u}$, the conjecture leads to the sequence $$1,1,1,2,3,5,8,12,18,27,39,55,\underline{78,108,150,207,284,388,532,726}$$ for primitive chord diagrams of degrees $m\le20$, with predictions underlined. The asymptotic behaviour $\lim_{m\to\infty}P_m/r^m= 1.06260548918755$ results, with $r=1.38027756909761$ solving $r^4=r^3+1$. Vassiliev invariants of knots are then …
We characterise the cabling operations on the weight systems of finite type knot invariants. The eigenvectors and eigenvalues of this family of operations are described over the cable eigenbasis. The action of immanent weight systems on general Feynman diagrams is considered, and the highest eigenvalue cabling eigenvectors are shown to be dual to the immanent weight systems. Using these results, we prove a recent conjecture of Bar-Natan and Garoufalidis on cablings of weight systems.
The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.
The general structure of the perturbative expansion of the vacuum expectation value of a Wilson line operator in Chern-Simons gauge field theory is analyzed. The expansion is organized according to the independent group structures that appear at each order. It is shown that the analysis is greatly simplified if the group factors are chosen in a certain way that we call canonical. This enables us to show that the logarithm of a polinomial knot invariant can be written in terms of primitive Vassiliev invariants only.
We compute the universal weight system for Vassiliev invariants coming from the Lie superalgebra gl(1|1) applying the construction of \cite{YB}. This weight system is a function from the space of chord diagrams to the center $Z$ of the universal enveloping algebra of gl(1|1), and we find a combinatorial expression for it in terms of the standard generators of $Z$. The resulting knot invariants generalize the Alexander-Conway polynomial.
The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function we show that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graph-complexes we show that this splitting on the level of the bialgebra of chord d…
In this note we reconsider a familiar result in Vassiliev knot theory - that the coefficients of the Alexander-Conway polynomial determine the top row of the Kontsevich integral - from the point of view of Kazuo Habiro's clasper theory. We observe that in this setting the calculation reflects the topology of the universal cyclic covering space of a claspered knot's complement.
We construct gauge invariant operators for singular knots in the context of Chern-Simons gauge theory. These new operators provide polynomial invariants and Vassiliev invariants for singular knots. As an application we present the form of the Kontsevich integral for the case of singular knots.
This paper contains a proof that chromatic weight systems, introduced by Chmutov, Duzhin and Lando, can be expressed in terms of weight systems associated with direct sums of the Lie algebras gl_n and so_n. As a consequence the Vassiliev invariants of knots corresponding to the chromatic weight systems distinguish exactly the same knots as a one variable specialisation Y of the Homfly and Kauffman polynomial.
We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-…
We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We also introduce singular semiquandles and virtual singular semiquandles which define invariants of flat singular virtual knots and links. As an application, we use semiquandle invariants to compare two Vassiliev invariants.
We show that (as conjectured by Lin and Wang) when a Vassiliev invariant of type $m$ is evaluated on a knot projection having $n$ crossings, the result is bounded by a constant times $n^m$. Thus the well known analogy between Vassiliev invariants and polynomials justifies (well, at least {\em explains}) the odd title of this note.
We determine the rational homology of the space of long knots in R^d for $d\geq4$. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E^1 page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with a bracket of degree d-1, which can be obtained as the homology of an explicit graph complex and is in theory completely computable. Our proof is a combination of a relative version of Kontsevich's formality of the little d-disks operad and of Sinha's cosimplicial …
We consider vector spaces H(n,l) and F(n,l) spanned by the degree-n coefficients in power series forms of the Homfly and Kauffman polynomials of links with l components. Generalizing previously known formulas, we determine the dimensions of the spaces H(n,l), F(n,l) and H(n,l)+F(n,l) for all values of n and l. Furthermore, we show that for knots the algebra generated by H(n,1)+F(n,1) (n > 0) is a polynomial algebra with dim(H(n,1)+F(n,1))-1=n+[n/2]-4 generators in degree n>3 and one generator in degrees 2 and 3.
This is a substantially revised version. The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null move leads in a natural way to the introduction of trivalent graphs with beads, and to a …
This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the perturbation expansion of the functional integral.
Consider the space of `long knots' in R^n, K_{n,1}. This is the space of knots as studied by V. Vassiliev. Based on previous work of the authors, it follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson algebra. A partial description of a basis is given here. In addition, the mod-p homology of this space is a `free, restricted Gerstenhaber-Poisson algebra'. Recursive application of this theorem allows us to deduce that there is p-torsion of all orders in the integral homology of K_{3,1}. This leads to some natural questions about the homotopy type of the space of long knot…
We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can…
Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge group. From this expansion new numerical knot invariants are obtained. These knot invariants turn out to be of finite type (Vassiliev invariants), and to possess an integral representation. Using known results about Jones, HOMFLY, Kauffman and Akutsu-Wadati polynomial invariants these new knot invariants are computed up to type six for all prime knots up to…
We show that a variation of Milnor's $\barμ$-invariants, the so-called Campbell-Hausdorff invariants introduced recently by Stefan Papadima, are of finite type with respect to {\it marked singular links}. These link invariants are stronger than quantum invariants in the sense that they detect easily the non-invertibility of links with more than one components. It is still open whether some effectively computable knot invariants, e.g. finite type knot invariants (Vassiliev invariants), could detect the non-invertibility of knots.
The general structure of the perturbative expansion of the vacuum expectation value of a product of Wilson-loop operators is analyzed in the context of Chern-Simons gauge theory. Wilson loops are opened into Wilson lines in order to unravel the algebraic structure encoded in the group factors of the perturbative series expansion. In the process a factorization theorem is proved for Wilson lines. Wilson lines are then closed back into Wilson loops and new link invariants of finite type are defined. Integral expressions for these invariants are presented for the first three primitive ones of low…
Homomorphic expansions are combinatorial invariants of knotted objects, which are universal in the sense that all finite-type (Vassiliev) invariants factor through them. Homomorphic expansions are also important as bridging objects between low-dimensional topology and quantum algebra. For example, homomorphic expansions of parenthesised braids are in one-to-one correspondence with Drinfel'd associators (Bar-Natan 1998), and homomorphic expansions of $w$-foams are in one-to-one correspondence with solutions to the Kashiwara-Vergne (KV) equations (Bar-Natan and the first author, 2017). The sets …
In this paper we compute a $q$-hypergeometric expression for the cyclotomic expansion of the colored Jones polynomial for the left-handed torus knot $(2,2t+1)$ and use this to define a family of quantum modular forms which are dual to the generalized Kontsevich-Zagier series.
