Polynomial algorithm for alternating link equivalence

TL;DR

This paper presents a polynomial-time algorithm for determining link equivalence up to isotopy specifically for alternating links, leveraging topological and graph-theoretic insights.

Contribution

It introduces the first polynomial algorithm for alternating link equivalence, utilizing Tait flyping conjectures and graph theory complexity results.

Findings

Algorithm runs in polynomial time based on maximum crossings

Uses Tait flyping conjectures and topological graph theory

Establishes complexity bounds for link equivalence problem

Abstract

Link equivalence up to isotopy in a 3-space is the problem that lies at the root of knot theory, and is important in 3-dimensional topology and geometry. We consider its restriction to alternating links, given by two alternating diagrams with $n_1$ and $n_2$ crossings, and show that this problem has polynomial algorithm in terms of $max\{n_1, n_2\}$. For the proof, we use Tait flyping conjectures, observations stemming from the work of Lackenby, Menasco, Sundberg and Thistlethwaite on alternating links, and algorithmic complexity of some problems from graph theory and topological graph theory.