An algorithm for comparing Legendrian knots

TL;DR

This paper presents an algorithm to determine the equivalence of Legendrian and transverse links, with practical improvements for links with small symmetry groups, enabling more efficient comparisons.

Contribution

The paper introduces a new algorithm for comparing Legendrian and transverse links, including optimizations for links with small symmetry groups.

Findings

Algorithm can decide link equivalence in theory.

Practical improvements enable comparison of many links.

Efficiency depends on the symmetry group size.

Abstract

We construct an algorithm to decide whether two given Legendrian or transverse links are equivalent. In general, the complexity of the algorithm is too high for practical implementation. However, in many cases, when the symmetry group of the link is small and explicitly known, the most time-consuming part of the algorithm can be bypassed, thus allowing one to compare many pairs of Legendrian and transverse links in practice.