arXiv research

The Vassiliev conjecture states that the Vassiliev invariants are dense in the space of all numerical link invariants in the sense that any link invariant is a pointwise limit of Vassiliev invariants. In this article, we prove that the Vassiliev conjecture holds in the case of the coefficients of the HOMFLY and the Kau…

We find approximations by Vassiliev invariants for the coefficients of the Jones polynomial and all specializations of the HOMFLY and Kauffman polynomials. Consequently, we obtain approximations of some other link invariants arising from the homology of branched covers of links.

We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant $V_2$, are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is appli…

New formulas for spatial 2-bouquet graphs discovered.

Enhances Vassiliev knot invariants using chord diagrams.

Paper introduces simplified formulas for Milnor's triple linking number.

Defines Vassiliev complexity measures for open and closed curves in 3D space.

Study Vassiliev invariants and periodic orbits of Axiom A flows.

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 inva…

As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants of a knot in an $R^1$-fibration over a surface $F$. They take values in the free…

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this …

We prove that if n\ge1, then an (n+1)-component Brunnian link L in a connected, oriented 3-manifold is C_n-equivalent to an unlink. We also prove that if n\ge2, then L can not be distinguished from an unlink by any Goussarov-Vassiliev finite type invariant of degree<2n.

Study on multiple linking numbers, extending Gauss diagram formulas.

In 1993 K. Habiro defined $C_k$-move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by $C_k$-moves if and only if they have the same Vassiliev invariants of order $\leq k-1$. In this paper we define Vassiliev invariant of type $(k_1,...,k_l)$, and show that, for $k=k…

Explains basic knot and link theory for students.

In this paper we introduce two theories of finite type invariants for framed links with fixed linking matrix. We show that these thepries are related to the theory of Vassiliev invariants of framed links. We also study the corresponding spaces of ``chord diagrams''.

We show that the coefficients of the re-normalized link invariants of the paper "Multivariable link invariants arising from Lie superalgebras of type I" are Vassiliev invariants which give rise to a canonical family of weight systems.

New skein theory for Links-Gould polynomial simplifies link evaluations.

We show that the Vassiliev invariants of orders $\leq n$ 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. As a consequence of this, we obtain that the Vassiliev invariants of …

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 pert…

Pulling back the weight system associated with the exceptional Lie algebra G_2 by a modification of the universal Vassiliev-Kontsevich invariant yields a link invariant; extending it to 3-nets, we derive a recursive algorithm for its evaluation.

We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for…

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…

We consider surgery moves along (n+1)-component Brunnian links in compact connected oriented 3-manifolds, where the framing of the each component is 1/k for k in Z. We show that no finite type invariant of degree < 2n-2 can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We…

Study algebraic relations of Vassiliev invariants for families of knots.

A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, the first author proved that the restriction to Brunnian links of any Goussarov-Vassiliev finite type invariant of (n+1)-component links of degree<2n is trivial. The purpose of this paper is to study the first nont…

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's $μ$ invariant. These invariants are seen to be naturally r…

We prove the existence of a degree 7 Vassiliev invariant of long (or string) two-component links which is not preserved under the simultaneous change of orientation of both components. The non-invertibility of this invariant can be detected by the standard weight system with values in the tensor square of the universal…

We introduce the concept of `claspers,' which are surfaces in 3-manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called `C_k-equivalence,' which is generated by surgery operations of a certain kin…

A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides …

We define a multi-variable version of the Affine Index Polynomial for virtual links. This invariant reduces to the original Affine Index Polynomial in the case of virtual knots, and also generalizes the version for compatible virtual links recently developed by L. Kauffman. We prove that this invariant is a Vassiliev i…

We find that Koschorke's $β$-invariant and the triple $μ$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin…

The purpose of the paper is twofold. First, we give a short proof using the Kontsevich integral for the fact that the restriction of an invariant of degree 2n to (n+1)-component Brunnian links can be expressed as a quadratic form on the Milnor mu-bar link-homotopy invariants of length n+1. Second, we describe the struc…

The usual construction of link invariants from quantum groups applied to the superalgebra D_{2 1,alpha} is shown to be trivial. One can modify this construction to get a two variable invariant. Unusually, this invariant is additive with respect to connected sum or disjoint union. This invariant contains an infinity of …

The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface $M\times I$ …

For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant …

The article calculates a multiplying factor to convert rational Vassiliev invariants to integer-valued ones.

New method computes first Vassiliev derivative of Khovanov homology.

In this paper we define and present a simple combinatorial formula for a 3-variable Laurent polynomial invariant of conjugacy classes in Artin braid group $B_m$. We show that this Laurent polynomial satisfies the Conway skein relation and its coefficients are Vassiliev invariants of braids.

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 $\leq k-1$. We see that the set of $C_k$-…

We give a criterion to detect whether the derivatives of the HOMFLY polynomial at a point is a Vassiliev invariant or not. In particular, for a complex number b we show that the derivative P_K^{(m,n)}(b,0)=d^m/da^m d^n/dx^n P_K(a,x)|(a, x) = (b, 0) of the HOMFLY polynomial of a knot K at (b,0) is a Vassiliev invariant …

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of complex-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact $R^3$ are canonically isomorphic. Recently we constructed the first examples where Vassiliev in…

The book defines an invariant for framed links and 3-manifolds using graph configurations.

In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, th…

The paper connects ADO polynomials to Vassiliev invariants for knots.

Study Vassiliev invariants for virtual knots, expanding quantum theory.

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

A formula for the difference of Vassiliev invariants of degree k+1 of two knots all of whose Vassiliev invariants of degree k agree is proven. The proof uses K. Habiro's C-moves and his theorem which relates them to Vassiliev invariants.