Vassiliev Invariants Research Articles

Delta move and polynomial invariants of links

We investigate how a self-delta move, which is a delta move on the same component, influences the HOMFLY polynomial of a link. Then we reveal some relationships among finite type invariants, which are coming from the derivatives of the Jones polynomials and the first HOMFLY coefficient polynomials, of the four links involving in a self-delta move.

UNIVERSAL VASSILIEV INVARIANTS OF LINKS IN COVERINGS OF 3-MANIFOLDS

We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M. We construct universal Vassiliev invariants of links in M, where M=P2×[0,1] is a cylinder over the real projective plane P2, M=Σ×[0,1] is a cylinder over a surface Σ with boundary, and M=S1×S2. A finite covering p:N→M induces a map π1(p)* between labeled chord diagrams that corresponds to taking the preimage p-1(L)⊂N of a link L⊂M. The maps p-1 and π1(p)* intertwine the constructed universal Vassiliev invariants.

Finite type invariants of cyclic branched covers

Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parameterized by the positive integers), namely the cyclic branched coverings of the knot. In this paper, we give a formula for the Casson–Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its signature function.

Vassiliev invariants and the Poincaré conjecture

This article examines the relationship between 3-manifold topology and knot invariants of finite type. We prove that in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. If, on the other hand, Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true (i.e. every homotopy 3-sphere is homeomorphic to the standard 3-sphere).

FINITE TYPE INVARIANTS FOR SINGULAR SURFACE BRAIDS ASSOCIATED WITH SIMPLE 1-HANDLE SURGERIES

S. Kamada introduced finite type invariants of knotted surfaces in 4-space associated with finger moves and 1-handle surgeries. In this paper, we define finite type invariants of surface braids associated with simple 1-handle surgeries and prove that a certain set of finite type invariants controls all finite type invariants. As a consequence, we see that every finite type invariant is not a complete invariant.

Gauß diagram sums on almost positive knots

Using the Fiedler–Polyak–Viro Gauß diagram formulae we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of a given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property P.

Braids on surfaces and finite type invariants

We prove that there is no functorial universal finite type invariant for braids in Σ× I if the genus of Σ is positive. To cite this article: P. Bellingeri, L. Funar, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Finite-type invariants for graphs and graph reconstructions

Many of the fundamental open problems in graph theory have the following general form: How much information does one need to know about a graph G in order to determine G uniquely. In this article we investigate a new approach to this sort of problem motivated by the notion of a finite-type invariant, recently introduced in the study of knots. We introduce the concepts of vertex-finite-type invariants of graphs, and edge-finite-type invariants of graphs, and show that these sets of functions have surprising algebraic properties. The study of these invariants is intimately related with the classical vertex- and edge-reconstruction conjectures, and we demonstrate that the algebraic properties of the finite-type invariants lead immediately to some of the fundamental results in graph reconstruction theory.

Local moves on spatial graphs and finite type invariants

We define A k -moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let A k -equivalence denote an equivalence relation generated by A k -moves and ambient isotopy. A k -equivalence implies A k-1 -equivalence. Let F be an A k-1 -equivalence class of the embeddings of a finite graph into the 3-sphere. Let g be the quotient set of F under A k -equivalence. We show that the set g forms an abelian group under a certain geometric operation. We define finite type invariants on F of order (n; k). And we show that if any finite type invariant of order (1; k) takes the same value on two elements of F, then they are A k -equivalent. A k -move is a generalization of C k -move defined by K. Habiro. Habiro showed that two oriented knots are the same up to C k -move and ambient isotopy if and only if any Vassiliev invariant of order < k - 1 takes the same value on them. The 'if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be C k -equivalent.

An Orientation-Sensitive Vassiliev Invariant for Virtual Knots

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.

Vassiliev invariants for braids on surfaces

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.

A geometric characterization of Vassiliev invariants

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree m if and only if it is a polynomial of degree m on every geometric sequence of knots. Here a sequence Kz with z 2 Z is called geometric if the knots Kz coincide outside a ball B, inside of which they satisfy Kz \B = z for all z and some pure braid . As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in S 1 ◊ S 2 that can be distinguished by Z/2-invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over Z a universal Vassiliev invariant of degree 1 for knots in S 1 ◊S 2 .

Grope cobordism of classical knots

Motivated by the lower central series of a group, we define the notion of a grope cobordism between two knots in a 3-manifold. Just like an iterated group commutator, each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov–Habiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants. The derived commutator series of a group also has a three-dimensional analogy, namely knots modulo symmetric grope cobordism. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the Cochran–Orr–Teichner filtration of the knot concordance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2 h ), Blanchfield forms or S-equivalence at h=2, Casson–Gordon invariants at h=3, and for h=4 one finds the new von Neumann signatures of a knot.

Enhanced Linking Numbers

1. INTRODUCTION. The study of knots and links begins with simple intuitive problems but quickly leads to sophisticated mathematics. This paper will provide the reader with an accessible route that begins with basic knot theory and leads into interesting realms of modern research. The specific topic of the paper is the enhanced linking number, X, a new invariant of links that provides a simple tool to address some of the fundamental problems in the study of linking. The linked rings illustrated on the left in Figure 1 are clearly linked; when a magician pulls such a link apart we know we have been tricked. A mathematical proof that this link is nontrivial depends on showing that its linking number is 1, where the linking number is the basic invariant of link theory, first described by Gauss two hundred years ago. With the link on the right the situation is less clear. If this link is built from rope or beads, then one quickly finds that unlinking the two is impossible. However, formulating a mathematical proof of this is more difficult; for instance, the linking number turns out to be of no help. The enhanced linking number reduces the proof that this link is nontrivial to a simple calculation. This article describes the enhanced linking number and shows how it can be applied to this basic problem of distinguishing links. It will also discuss the role of X in a variety of advanced topics, such as periodicity, symmetry, Brunnian linking, and the modern theory of finite type invariants.

ON A MAP FROM PURE BRAIDS TO KNOTS

We introduce an oriented version of the plat closure and prove a Markov-type theorem for it. Some implications for Vassiliev invariants are discussed.

Milnor numbers, spanning trees, and the Alexander–Conway polynomial

We study relations between the Alexander–Conway polynomial ∇ L and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of ∇ L of an m-component link L all of whose Milnor numbers μ i 1… i p vanish for p⩽ n. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.

Ck-moves on spatial theta-curves and Vassiliev invariants

The C k -equivalence is an equivalence relation generated by C k -moves defined by Habiro. Habiro showed that the set of C k -equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order ⩽ k−1. We see that the set of C k -equivalence classes of the spatial θ-curves forms a group under the vertex connected sum and that if the group is abelian, then it can be classified by the additive Vassiliev invariant of order ⩽ k−1. However the group is not necessarily abelian. In fact, we show that it is nonabelian for k⩾12. As an easy consequence, we have the set of C k -equivalence classes of m-string links, which forms a group under the composition, is nonabelian for k⩾12 and m⩾2.

Two applications of elementary knot theory to Lie algebras and Vassiliev invariants

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 computation of 1+1=2 on an abacus. The Wheels conjecture is proved from the fact that the k-fold connected cover of the unknot is the unknot for all k. Along the way, we find a formula for the invariant of the general (k,l) cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo-Kirillov map S(g)-->U(g) for metrized Lie (super-)algebras g.

Isomorphism of the Groups of Vassiliev Invariants of Legendrian and Pseudo-Legendrian Knots

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of C-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples of contact 3-manifolds where Vassiliev invariants of Legendrian and of framed knots are different. Moreover in these examples Vassiliev invariants of Legendrian knots distinguish Legendrian knots that are isotopic as framed knots and homotopic as Legendrian immersions. This raised the question what information about Legendrian knots can be captured using Vassiliev invariants. Here we answer this question by showing that for any contact 3-manifold with a cooriented contact structure the groups of Vassiliev invariants of Legendrian knots and of knots that are nowhere tangent to a vector field that coorients the contact structure are canonically isomorphic.

Finite-Type Invariants of Cubic Complexes

The paper is for a general audience and may serve as a preliminary introduction to the theory of finite-type invariants.