Special alternating links of minimal unlinking number

TL;DR

This paper investigates the unlinking number of special alternating links, establishing conditions under which the classical signature bound is sharp and demonstrating its implications for computing unknotting numbers of certain knots.

Contribution

It proves that for special alternating links, the signature bound on the unlinking number is always realized in any alternating diagram, and applies this to compute unknotting numbers for specific knots.

Findings

Signature bound is sharp for certain special alternating links.

Unlinking number is realized by crossing changes in any alternating diagram.

New unknotting numbers for some knots with crossing number 11 and 12.

Abstract

For any link in the 3-sphere, there is a natural lower bound for the unlinking number in terms of the classical signature. We prove that if this lower bound is sharp for a special alternating link $L$, then the unlinking number of $L$ is necessarily realized by crossing changes in any alternating diagram for $L$. As an application, we compute new values of the unknotting numbers for some special alternating knots with crossing number 11 and 12.