Virtual Knot Groups

TL;DR

This paper introduces new quotient groups associated with knot diagrams that extend classical knot groups, maintaining invariance under Reidemeister moves and generalizing previous extended knot groups.

Contribution

It defines a set of quotient groups for knot diagrams that unify and extend existing extended knot groups while preserving invariance under Reidemeister moves.

Findings

The quotient groups are invariant under Reidemeister moves.

The set includes previously defined extended knot groups.

The framework generalizes classical knot groups.

Abstract

For a knot diagram $K$, the classical knot group $\pi_1(K)$ is a free group modulo relations determined by Wirtinger-type relations on the classical crossings. The classical knot group is invariant under the Reidemeister moves. In this paper, we define a set of quotient groups associated to a knot diagram $K$. These quotient groups are invariant under the Reidemeister moves and the set includes the extended knot groups defined by Boden et al and Silver and Williams.