Equivariant knots and knot Floer homology

TL;DR

This paper introduces equivariant concordance invariants derived from knot Floer homology, demonstrating their effectiveness in bounding equivariant slice genus and detecting exotic slice disks, with implications for the structure of the equivariant concordance group.

Contribution

The authors define new equivariant invariants from knot Floer homology and apply them to answer open questions about equivariant slice genus and exotic slice disks.

Findings

Equivariant invariants provide lower bounds for equivariant slice genus.

Construction of strongly invertible slice knots with arbitrarily large equivariant slice genus.

Detection of exotic pairs of slice disks using knot Floer homology.

Abstract

We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly non-equivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden, and extend a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route towards establishing the non-commutativity of the equivariant concordance group.