Unknotting and Ascending Numbers of Knots and their Families

TL;DR

This paper calculates ascending numbers for knots with up to 10 crossings, proves a signature theorem for alternating knot families, and derives formulas for unknotting and ascending numbers for specific knot families.

Contribution

It introduces new formulas for ascending numbers of certain knot families and confirms unknotting numbers using signature-based methods.

Findings

Ascending numbers determined for 64 knots with up to 10 crossings

Signature theorem for alternating knot families proved

Formulas derived for ascending numbers of 11 knot families

Abstract

Ascending numbers are determined for 64 knots with at most n=10 crossings. After proving the theorem about the signature of alternating knot families, we distinguished all families of knots obtained from generating alternating knots with at most 10 crossings, for which the unknotting number can be confirmed by using the general formulae for signatures. For 11 families of knots general formulae are obtained for their ascending numbers.