nLab Vassiliev skein relation



Knot theory


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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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.

[[!include SVG skein double crossing]] [[!include SVG skein positive crossing]] [[!include SVG skein negative crossing]] L d L + L \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}


General discussion includes

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

category: knot theory

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