nLab Vassiliev skein relation

Contents

Context

Knot theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

The Vassiliev skein relation is a way to extend knot invariants to singular knots (at least, to singular knots where the only singularities are double points). If vv is a knot invariant that takes values in an abelian group, then it is extended to singular knots using the relation

v(L d)=v(L +)v(L ) v(L_d) = v(L_+) - v(L_-)

where L dL_d is a singular knot with a double point and L +L_+, respectively L L_-, are formed from L dL_d by replacing the double point by a positively oriented, respectively negatively oriented, crossing.

References

General discussion:

Discussion in the context of quantization of 3d Chern-Simons theory:

category: knot theory

Last revised on September 10, 2022 at 03:11:19. See the history of this page for a list of all contributions to it.