Narrow Results

We review quantum field theory approach to the knot theory. Using holomorphic gauge, we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.

We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and the loop expansion through Vassiliev invariants explicitly demonstrate this phenomenon. We study the genus expansion and discuss its properties. We also consider the loop expansion in details. In particular, we give an algorithm to calculate Vassiliev invariants, give some examples and discuss relations among Vassiliev invariants. Then we consider superpolynomials for torus knots defined via double affine Hecke algebra. We claim that the superpolynomials are not functions of Hurwitz type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are beta-deformed to Hamiltonians of the Calogero–Moser–Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials.

We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on $g+1$ parameters $n_1,\dots ,n_{g+1}$. These invariants are symmetric polynomials in $n_1,\dots ,n_{g+1}$ whose degree coincide with their order. We also discuss their topological and integer-valued properties.

We introduce a new approach to Vassiliev invariants. This approach deals with Vassiliev invariants directly on knots and does not make use of diagrams. We give a series of applications of this approach, (re)proving some new and known facts on Vassiliev invariants.

It is shown by Y. Ohyama, K. Taniyama and S. Yamada that for any natural number n and any knot K, there are infinitely many unknotting number one knots, whose Vassiliev invariants of order less than or equal to n coincide with those of K. We analyze it for delta unknotting number, and obtain the following. For any natural number n and any oriented knots K and M with a₂(K)≠a₂(M) there are infinitely many knots J_m such that the delta distance between J_m and M coincide with |a₂(K)-a₂(M)| and whose Vassiliev invariants of order less than or equal to n coincide with those of K. Here a₂(K) is the second coefficient of the Conway polynomial of K.

In [1], E. Appleboim introduced the notion of double dating linking class-P invariants of finite type for framed links with a fixed linking matrix P and showed that all Vassiliev link invariants are of finite type for any linking matrix and in [13], R. Trapp provided a necessary condition for a knot invariant to be a Vassiliev invariant by using twist sequences. In this paper we provide a necessary condition for a framed link invariant to be a DD-linking class-P invariant of finite type by considering sequence of links induced from a double dating tangle. As applications we give a generalization of R. Trapp's result to see whether a link invariant is a Vassiliev invariant or not and apply the criterion for all non-zero coefficients of the Jones, HOMFLY, Q-, and Alexander polynomial.

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal Vassiliev link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel anomaly -that groups the Bott and Taubes anomalous terms- is a combination of diagrams with two univalent vertices and we explicitly define the isomorphism of which transforms the Kontsevich integral into the Poirier limit of the perturbative expression of the Chern-Simons theory for framed links, as a function of α

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.

The transforms of two oriented parallel strands to a k-half twist of two strands are called t_k-move and -move respectively depending on the orientations of the two strands. In this paper we give criterions to detect whether a knot K can be transformed to a knot K' by t_2k-moves and -moves respectively and if so, we give some results on how many moves are needed in these transformations respectively, by using some Vassiliev invariants. Moreover we give a relation between the Δ-move and the t_2k-move by considering the coefficient of z² in the Conway polynomial of a knot, which is a Vassiliev invariant of degree 2.

In this paper, we study lens knots and periodic knots by using integral Vassiliev invariants. Knot polynomials such as the Jones, HOMFLY, Kauffman polynomials give infinitely many integral Vassiliev invariants and we get some necessary conditions for a link to be a lens knot or a periodic link by using these polynomial invariants.

We define algebraic structures on graph cohomology and prove that they correspond to algebraic structures on the cohomology of the spaces of imbeddings of S¹ or ℝ into ℝⁿ. As a corollary, we deduce the existence of an infinite number of nontrivial cohomology classes in Imb(S¹, ℝⁿ) when n is even and greater than 3. Finally, we give a new interpretation of the anomaly term for the Vassiliev invariants in ℝ³.

Let s_Q be the satellite operation on knots defined by a pattern (V, Q), where V is a standard solid torus in S³ and Q ⊂ V is a knot that is geometrically essential in V. It is known (Kuperberg [5]) that if v is any knot invariant of order n ≥ 0, then v ◦ s_Q is also a knot invariant of order ≤ n. We show that if the knot Q has the winding number zero in V, then the satellite map passes n-equivalent knots into (n + 1)-equivalent ones. Kalfagianni [4] has defined for each nonnegative integer n surgery n-trivial knots and studied their properties. It is known that for each n every surgery n-trivial knot is n-trivial. We show that for each n there are n-trivial knots which do not admit a non-unitary n-trivializer that show they to be surgery n-trivial. Przytycki showed [12] that if a knot Q is trivial in S³ and is embedded in V in such a way that it is k-trivial inside V and if a knot is m-trivial, then the satellite knot is (k + m + 1)-trivial. We establish a version of aforementioned Przytycki's result for surgery n-triviality, refining thus a construction for surgery n-trivial knots suggested by Kalfagianni.

Whether Vassiliev invariants can distinguish all knots or not is a well-known open problem which is equivalent to the question whether the similarity index of any two different knots is finite or not. We give relations between the degrees of Vassiliev invariants and the similarity indices of knots, links and tangles. From these, we get necessary conditions for a knot invariant to be a Vassiliev invariant and get methods to detect the similarity index of two knots or tangles.

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 Kauffman polynomials.

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. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

Whether Vassiliev invariants can distinguish all knots or not is a well-known open problem which is equivalent to the question whether the similarity index of any two different knots is finite or not.

Let T and S be two tangles which are n-similar for some natural number n and let the closure of T be well-defined. Let T* and S* be the mirror images of T and S respectively. Then we show that for any prime number p, mod p for any integral Vassiliev invariant v of degree ≤ np. We also show that for any Vassiliev invariant w of degree ≤ n if n is odd. Therefore, if an amphicheiral knot can be distinguished from a trivial knot by a Vassiliev invariant, then it has an even triviality index.

From these, we get some necessary conditions for a knot invariant to be a Vassiliev invariant and get a method to detect the similarity index of two knots or tangles.

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 is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular flat virtual knots with one double-point.

In this paper we study the ℤ-module A₂ of two-chord diagrams for knots with zero winding number in the solid torus KST₀, which is needed in studying the type-two invariants for knots in KST₀. We show that this module (or abelian group), which is given as a presentation with infinite number of generators and an infinite number of relations, is a free infinitely generated module. Moreover, we show that this module is isomorphic to the direct sum of three free modules that are easier to understand.

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a type 2 vassiliev invariant coming from the polynomial.

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math.191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.