Integer inequalities between knot invariants, skein tree depth and delta-crossing numbers

TL;DR

This paper investigates the complexity of computing knot polynomials through skein trees, establishing new bounds on skein tree depth and exploring inequalities among knot invariants, supported by computational data and visualizations.

Contribution

It introduces new upper and lower bounds on skein tree depth and provides tables for knots and links with previously undetermined skein depths, combining theoretical and computational approaches.

Findings

New upper bound on skein tree depth for links

New lower bound on skein tree depth

Tables of knots with undetermined skein depths

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 the HOMFLY-PT polynomial). We combine theoretical and computational results on the skein tree depth of knots and links. 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). The paper surveys known (and new) inequalities between integer-valued classical knot invariants. It features a visual graph of the relations.