Rewriting Systems in Alternating Knot groups with the Dehn presentation

TL;DR

This paper proves that the Dehn presentation of the knot group for tame, prime, alternating knots admits a finite, complete, and terminating rewriting system, facilitating algebraic manipulations of knot groups.

Contribution

It introduces a finite, complete rewriting system for the Dehn presentation of knot groups of tame, prime, alternating knots, ensuring termination despite some rules increasing word length.

Findings

The rewriting system is finite and complete.

The system is terminating despite length-increasing rules.

Applicable to knots with a common regular projection.

Abstract

Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the Dehn presentation of the knot group of a tame, prime, alternating knot, with a regular and common projection has a finite and complete rewriting system. Although there are rules in the rewriting system with left-hand side a generator and which increase the length of the words we show that the system is terminating.