# nLab Vassiliev skein relation

### Context

#### Topology

topology

algebraic topology

## Examples

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

$v\left({L}_{d}\right)=v\left({L}_{+}\right)-v\left({L}_{-}\right)$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.

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4.0 0.0 C 4.0 2.20917 2.20917 4.0 0.0 4.0 C -2.20917 4.0 -4.0 2.20917 -4.0 0.0 C -4.0 -2.20917 -2.20917 -4.0 0.0 -4.0 C 2.20917 -4.0 4.0 -2.20917 4.0 0.0 Z M 0.0 0.0 " style="stroke:none"/> </g> </g> </g> </g> </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="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 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}

category: knot theory

Revised on April 1, 2011 09:22:55 by Andrew Stacey (129.241.15.200)