Equivariantly slicing strongly negative amphichiral knots

TL;DR

This paper develops new obstructions based on algebraic and gauge-theoretic invariants to determine when strongly negative amphichiral knots can be slice in an equivariant manner, advancing understanding of knot symmetry and sliceness.

Contribution

It introduces novel obstructions using determinant, Spinc-structures, Donaldson's theorem, and Heegaard Floer correction terms for equivariant sliceness of strongly negative amphichiral knots.

Findings

Out of 16 such knots with ≤12 crossings, 8 are not equivariantly slice.

Constructed equivariant ribbon diagrams for 5 knots.

Provided new obstructions based on Heegaard Floer invariants.

Abstract

We prove obstructions to a strongly negative amphichiral knot bounding an equivariant slice disk in the 4-ball using the determinant, Spinc-structures and Donaldson's theorem. Of the 16 slice strongly negative amphichiral knots with 12 or fewer crossings, our obstructions show that 8 are not equivariantly slice, we exhibit equivariant ribbon diagrams for 5 others, and the remaining 3 are unknown. Finally, we give an obstruction to a knot being strongly negative amphichiral in terms of Heegaard Floer correction terms.