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 Vassiliev invariants for links from Chern-Simons perturbation theory

1997 ◽  
Vol 488 (3) ◽  
pp. 677-718 ◽  
Author(s):  
M. Alvarez ◽  
J.M.F. Labastida ◽  
E. Pérez
Keyword(s):  
Perturbation Theory ◽  
Vassiliev Invariants ◽  
Chern Simons

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.

scholarly journals PERTURBATIVE CHERN-SIMONS THEORY

1995 ◽  
Vol 04 (04) ◽  
pp. 503-547 ◽  
Author(s):  
DROR BAR-NATAN
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Integral Formula ◽  
Simons Theory ◽  
Gauge Field Theory ◽  
Complete Agreement ◽  
Knot Invariants ◽  
Arf Invariant ◽  
Chern Simons ◽  
Chern Simons Theory

We present the perturbation theory of the Chern-Simons gauge field theory and prove that to second order it indeed gives knot invariants. We identify these invariants and show that in fact we get a previously unknown integral formula for the Arf invariant of a knot, in complete agreement with earlier non-perturbative results of Witten. We outline our expectations for the behavior of the theory beyond two loops.

MANY CLASSICAL KNOT INVARIANTS ARE NOT VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (01) ◽  
pp. 7-10 ◽  
Author(s):  
JOHN DEAN
Keyword(s):  
Crossing Number ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Bridge Number

We show that under twisting, a Vassiliev invariant of order n behaves like a polynomial of degree at most n. This greatly restricts the values that a Vassiliev invariant can take, for example, on the (2, m) torus knots. In particular, this implies that many classical numerical knot invariants such as the signature, genus, bridge number, crossing number, and unknotting number are not Vassiliev invariants.

Vassiliev invariants and the Hopf algebra of chord diagrams

1996 ◽  
Vol 119 (1) ◽  
pp. 55-65 ◽  
Author(s):  
Simon Willerton
Keyword(s):  
Hopf Algebra ◽  
Canonical Projection ◽  
Double Points ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Type Formula ◽  
Chord Diagrams ◽  
Multivariate Calculus

AbstractThis paper is closely related to Bar-Natan's work, and fills in some of the gaps in [1]. Following his analogy of the extension of knot invariants to knots with double points to the notion of multivariate calculus on polynomials, we introduce a new notation which facilitates the formulation of a Leibniz type formula for the product of two Vassiliev invariants. This leads us to see how Bar-Natan's co-product of chord diagrams corresponds to multiplication of Vassiliev invariants. We also include a proof that the multiplication in is a consequence of Bar-Natan's 4T relation.The last part of this paper consists of a proof that the space of weight systems is a sub-Hopf algebra of the space *, by means of the canonical projection.

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