Higher-order Analogues of the Slice Genus of a Knot

TL;DR

This paper introduces higher-order genera as refined geometric invariants of knots, extending the slice genus concept, and establishes lower bounds using von Neumann $ ho$-invariants, enriching the understanding of knot concordance.

Contribution

It defines higher-order genera for knots, relates them to von Neumann $ ho$-invariants, and refines the Grope filtration of the knot concordance group.

Findings

Higher-order genera are well-defined for certain knots.

Lower bounds for these invariants are established via higher-order signatures.

The invariants provide a finer classification within the knot concordance group.

Abstract

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $\rho$-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group.