On a computation of the skein tree depth of knots and links

TL;DR

This paper introduces new bounds on the skein tree depth of knots and links, improving understanding of the complexity involved in computing link polynomials and providing new data on specific knots and links.

Contribution

It presents the first improved upper and lower bounds on skein tree depth and supplies tables for previously undetermined cases, advancing knot complexity analysis.

Findings

New upper bound on skein tree depth for links

New lower bound on skein tree depth for links

Tables of skein tree depths for various knots and links

Abstract

The maximum length of the shortest path from a leaf to the root of a skein tree for knots and links gives a measure of the complexity of computing link polynomials by the skein relation (the Jones polynomial, the Alexander-Conway polynomial, and more generally HOMFLY-PT polynomial). In this paper, we prove the new upper bound on the skein tree depth of a link and give examples of links where the new bound is stronger than the known bound. We also give the new lower bound. Moreover, we derive tables of knots and links with their skein tree depth that were up to now undetermined (for some of them, we give their range of possible values).