Signature invariants related to the unknotting number

TL;DR

This paper introduces improved lower bounds on the unknotting number of knots using signature functions, surpassing previous bounds and providing new insights into knot invariants and Gordian distances.

Contribution

It develops stronger signature-based bounds on the unknotting number, including for slice knots, and relates these to four-dimensional knot invariants and double-sliceness.

Findings

New bounds can be twice as strong as previous signature bounds

Bounds outperform those from Heegaard Floer and Khovanov homology

Results include bounds on Gordian distance and properties of slice knots

Abstract

New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordian distance between knots and information about four-dimensional knot invariants. By considering a related non-balanced signature function, bounds on the unknotting number of slice knots are constructed; these are related to the property of double-sliceness.