Equivariant algebraic concordance of strongly invertible knots

TL;DR

This paper introduces a new algebraic framework for strongly invertible knots, defining an invariant homomorphism that refines existing invariants and provides new obstructions to equivariant sliceness and bounds on slice genus.

Contribution

It constructs a homomorphism from the equivariant concordance group to a new algebraic group, extending previous invariants and introducing novel equivariant signatures and sliceness obstructions.

Findings

Defined a homomorphism $\

Provided new obstructions to equivariant sliceness.

Established lower bounds on equivariant slice genus.

Abstract

By considering a particular type of invariant Seifert surfaces we define a homomorphism $\Phi$ from the (topological) equivariant concordance group of directed strongly invertible knots $\widetilde{\mathcal{C}}$ to a new equivariant algebraic concordance group $\widetilde{\mathcal{G}}^{\mathbb{Z}}$. We prove that $\Phi$ lifts both Miller and Powell's equivariant algebraic concordance homomorphism, and Alfieri and Boyle's equivariant signature. Moreover, we provide a partial result on the isomorphism type of $\widetilde{\mathcal{G}}^{\mathbb{Z}}$, and we obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox-Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that $\Phi$ can obstruct equivariant sliceness for knots with Alexander polynomial one.