scholarly journals Kontsevich’s integral for the Kauffman polynomial

1996 ◽  
Vol 142 ◽  
pp. 39-65 ◽  
Author(s):  
Thang Tu Quoc Le ◽  
Jun Murakami
Keyword(s):  
Finite Type ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Knot Invariant ◽  
Kauffman Polynomial ◽  
Zamolodchikov Equation

Kontsevich’s integral is a knot invariant which contains in itself all knot invariants of finite type, or Vassiliev’s invariants. The value of this integral lies in an algebra A0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11].

scholarly journals INTERSECTION GRAPHS FOR STRING LINKS

2006 ◽  
Vol 15 (01) ◽  
pp. 53-72 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Adjacency Matrix ◽  
Intersection Graphs ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
String Links ◽  
Homfly Polynomials

We extend the notion of intersection graphs for chord diagrams in the theory of finite type knot invariants to chord diagrams for string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graphs, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.

scholarly journals Crosscap number of knots and volume bounds

2020 ◽  
Vol 31 (13) ◽  
pp. 2050111
Author(s):  
Noboru Ito ◽  
Yusuke Takimura
Keyword(s):  
Jones Polynomial ◽  
Upper Bounds ◽  
Crossing Number ◽  
Proved Theorem ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Hyperbolic Volume ◽  
Knot Invariant ◽  
Twist Number

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

COEFFICIENTS OF HOMFLY POLYNOMIAL AND KAUFFMAN POLYNOMIAL ARE NOT FINITE TYPE INVARIANTS

2002 ◽  
Vol 11 (04) ◽  
pp. 545-553 ◽  
Author(s):  
GYO TAEK JIN ◽  
JUNG HOON LEE
Keyword(s):  
Finite Type ◽  
Homfly Polynomial ◽  
Knot Invariants ◽  
Finite Type Invariants ◽  
Kauffman Polynomial

We show that the integer-valued knot invariants appearing as the nontrivial coefficients of the HOMFLY polynomial, the Kauffman polynomial and the Q-polynomial are not of finite type.

ON FINITE TYPE 3-MANIFOLD INVARIANTS I

1996 ◽  
Vol 05 (04) ◽  
pp. 441-461 ◽  
Author(s):  
STAVROS GAROUFALIDIS
Keyword(s):  
Vector Space ◽  
Finite Type ◽  
Commutative Algebra ◽  
Integral Homology ◽  
Knot Invariants ◽  
Finite Type Invariants ◽  
Definition Of ◽  
Type 3

Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3-spheres. In the present paper we propose another definition of finite type invariants of integral homology 3-spheres and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra. We compare the two induced filtrations on the vector space on the set of integral homology 3-spheres. As an observation, we discover a new set of restrictions that finite type invariants in the sense of Ohtsuki satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3-manifold invariants with the (Vassiliev) knot invariants.

Linearized chord diagrams and an upper bound for vassiliev invariants

2000 ◽  
Vol 09 (07) ◽  
pp. 847-853 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Upper Bound ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Chord Diagrams

Recently, Stoimenow [J. Knot Th. Ram. 7 (1998), 93–114] gave an upper bound on the dimension dn of the space of order n Vassiliev knot invariants, by considering chord diagrams of a certain type. We present a simpler argument which gives a better bound on the number of these chord diagrams, and hence on dn.

On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials

2007 ◽  
Vol 59 (2) ◽  
pp. 418-448 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Open Problem ◽  
Negative Answer ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Knot Invariant ◽  
Homfly Polynomials

AbstractIt is known that the Brandt–Lickorish–Millett–Ho polynomial Q contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from Q is an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree d ≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

scholarly journals Bott–Taubes–Vassiliev cohomology classes by cut-and-paste topology

2019 ◽  
Vol 30 (10) ◽  
pp. 1950047
Author(s):  
Robin Koytcheff
Keyword(s):  
Finite Type ◽  
Integer Lattice ◽  
Configuration Spaces ◽  
Euclidean Spaces ◽  
The Real ◽  
Knot Invariants ◽  
Trivalent Graph ◽  
Higher Dimensional ◽  
Valued Graph ◽  
Knots And Links

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct “Vassiliev classes” in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in [Formula: see text] for [Formula: see text]. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod-[Formula: see text] classes out of mod-[Formula: see text] graph cocycles, which need not be reductions of classes over the integers.

TWIST SEQUENCES AND VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (03) ◽  
pp. 391-405 ◽  
Author(s):  
ROLLAND TRAPP
Keyword(s):  
Finite Type ◽  
Polynomial Growth ◽  
Difference Sequence ◽  
Uniform Limit ◽  
Topological Information ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Vassiliev Invariant

In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.

scholarly journals Cabling the Vassiliev Invariants

1997 ◽  
Vol 06 (03) ◽  
pp. 327-358 ◽  
Author(s):  
A. Kricker ◽  
B. Spence ◽  
I. Aitchison
Keyword(s):  
Finite Type ◽  
Feynman Diagrams ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Eigenvectors And Eigenvalues

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. The canonical deframing projection for these knot invariants is 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.

A PROOF OF THE MELVIN–MORTON CONJECTURE AND FEYNMAN DIAGRAMS

1998 ◽  
Vol 07 (01) ◽  
pp. 23-40 ◽  
Author(s):  
S. CHMUTOV
Keyword(s):  
Hopf Algebra ◽  
Feynman Diagram ◽  
Invariant Function ◽  
Feynman Diagrams ◽  
Chord Diagrams ◽  
Knot Invariant ◽  
Difficult Part

The Melvin–Morton conjecture says how the Alexander–Conway knot invariant function can be read from the coloured Jones function. It has been proved by D. Bar-Natan and S. Garoufalidis. They reduced the conjecture to a statement about weight systems. The proof of the latter is the most difficult part of their paper. We give a new proof of the statement based on the Feynman diagram description of the primitive space of the Hopf algebra [Formula: see text] of chord diagrams.

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