Algorithmic simplification of knot diagrams: new moves and experiments

TL;DR

This paper introduces new moves for classical knot diagrams that enable monotonic, complete simplification in numerous tested examples, aiding in the analysis of knot complexity without increasing crossings.

Contribution

The paper presents a set of generalized moves extending Reidemeister moves, facilitating monotonic simplification of knot diagrams through experimental validation.

Findings

Moves successfully simplify diagrams in tested cases

Implementation available online for public use

No new theorems, focus on experimental methods

Abstract

This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diagram D' with the minimal possible number of crossings for the isotopy class of the knot represented by D. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain Z1, Z2, Z3 generalizing the classical Reidemeister moves R1, R2, R3, and another one C (together with a variant) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones. We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web.