Special alternating knots with sufficiently many twist regions have no chirally cosmetic surgeries

TL;DR

This paper proves that certain special alternating knots with many twist regions do not admit chirally cosmetic surgeries, and also establishes optimal bounds for specific finite type invariants.

Contribution

It demonstrates the non-existence of chirally cosmetic surgeries for large special alternating knots and solves Willerton's conjecture on finite type invariants.

Findings

Special alternating knots with more than 63 twist regions have no chirally cosmetic surgeries.

Provides optimal upper bounds for primitive finite type invariants of degree 2 and 3.

Solves Willerton's conjecture on finite type invariants.

Abstract

We show that a special alternating knot with sufficiently large number (more than $63$) of twist regions has no chirally cosmetic surgeries, a pair of Dehn surgeries producing orientation-reversingly homeomorphic $3$-manifolds. In the course of proof, we provide the optimal upper bounds of the primitive finite type invariants of degree 2 and 3 that solve Willerton's conjecture.