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.

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.

Weights of Feynman diagrams, link polynomials and Vassiliev knot invariants

1995 ◽  
Vol 04 (01) ◽  
pp. 163-188 ◽  
Author(s):  
Sergey Piunikhin
Keyword(s):  
Perturbation Theory ◽  
Irreducible Representation ◽  
Hopf Algebra ◽  
Lie Algebras ◽  
Feynman Diagrams ◽  
Primitive Elements ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Chern Simons

We prove that the construction of Vassiliev invariants by expanding the link polynomials Pg,V(q, q−1) at the point q=1 is equivalent to the construction of Vassiliev invariants from Chern-Simons perturbation theory. In both constructions a simple Lie algebra g and an irreducible representation V of g should be specified. We give an example of a Vassiliev invariant of order six which cannot be obtained by these constructions if we restrict ourselves to simple Lie algebras and do not allow semisimple ones. The explicit description of primitive elements in the Kontsevich Hopf algebra is given.

scholarly journals FINITE-TYPE KNOT INVARIANTS BASED ON THE BAND-PASS AND DOUBLED-DELTA MOVES

2010 ◽  
Vol 19 (03) ◽  
pp. 355-384 ◽  
Author(s):  
JAMES CONANT ◽  
JACOB MOSTOVOY ◽  
TED STANFORD
Keyword(s):  
Abelian Group ◽  
Large Class ◽  
Finite Type ◽  
Equivalence Classes ◽  
Connected Sum ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Conway Polynomial ◽  
Band Pass

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-type invariants, but things are more interesting for the doubled-delta move. The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander–Conway polynomial. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is ℤ2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n + 1 part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree 2n invariant.

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.

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].

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.

Knot Theory and Statistical Mechanics

1997 ◽  
Vol 11 (01n02) ◽  
pp. 39-49 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Statistical Mechanics ◽  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Scattering Amplitude ◽  
Basic Structure ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Quantum Link

This paper gives a self-contained exposition of the basic structure of quantum link invariants as state summations for a vacuum-vacuum scattering amplitude. Models of Vaughan Jones are expressed in this context. A simple proof is given that an important subset of these invariants are built from Vassiliev invariants of finite type.

scholarly journals PARITY AND EXOTIC COMBINATORIAL FORMULAE FOR FINITE-TYPE INVARIANTS OF VIRTUAL KNOTS

2012 ◽  
Vol 21 (13) ◽  
pp. 1240001 ◽  
Author(s):  
MICAH WHITNEY CHRISMAN ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Finite Type ◽  
Combinatorial Formula ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Additional Information ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Gauss Diagram

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer-valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov–Polyak–Viro finite-type. Moreover, every homogeneous Polyak–Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is non-constant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.

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