The rational Witt class and the unknotting number of a knot

TL;DR

This paper employs the rational Witt class of knots to derive new lower bounds and explicit values for the unknotting number, especially focusing on knots with low crossing numbers and pretzel knots.

Contribution

It introduces a novel application of the rational Witt class to estimate unknotting numbers, providing new bounds and exact values for specific classes of knots.

Findings

New lower bounds for unknotting numbers of low crossing knots

Explicit unknotting number values for some knots

Insights into unknotting numbers of pretzel knots

Abstract

We use the rational Witt class of a knot in the 3-sphere as a tool for addressing questions about its unknotting number. We apply these tools to several low crossing knots (151 knots with 11 crossing and 100 knots with 12 crossings) and to the family of n-stranded pretzel knots for various values of n>2. In many cases we obtain new lower bounds and in some cases explicit values for their unknotting numbers. Our results are mainly concerned with unknotting number one but we also address, somewhat more marginally, the case of higher unknotting numbers.