nLab
Vassiliev skein relation

Context

Knot theory

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic 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.

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fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0l57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0l57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \begin{svg}<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0l57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0l57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0l-57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0l-57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 1.7 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} \\ L_d & L_+ & L_- \end{array}

References

General discussion includes

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

category: knot theory

Revised on July 18, 2015 10:33:30 by Urs Schreiber (94.118.161.90)