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

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

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 $v$ is a knot invariant that takes values in an abelian group, then it is extended to singular knots using the relation

$v(L_d) = v(L_+) - v(L_-)$

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

$\begin{array}{ccc}
\begin{svg}[[!include SVG skein double crossing]]\end{svg} &
\begin{svg}[[!include SVG skein positive crossing]]\end{svg} &
\begin{svg}[[!include SVG skein negative crossing]]\end{svg} \\
L_d & L_+ & L_-
\end{array}$

References
General discussion includes

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

Last revised on July 18, 2015 at 10:33:30.
See the history of this page for a list of all contributions to it.