Instantons and some concordance invariants of knots

TL;DR

This paper introduces new knot concordance invariants derived from instanton homology with local coefficients, which can potentially bound the genus and double points of immersed surfaces bounded by the knot.

Contribution

The authors define a family of concordance invariants from instanton homology that extend previous work and offer new tools for studying knot concordance.

Findings

Defined a 1-parameter family of homomorphisms $f_{r}$ from the knot concordance group to the reals.

These invariants can potentially bound the genus of surfaces bounded by knots.

The invariants provide new insights into knot concordance and surface genus bounds.

Abstract

Concordance invariants of knots are derived from the instanton homology groups with local coefficients, as introduced in earlier work of the authors. These concordance invariants include a 1-parameter family of homomorphisms $f_{r}$, from the knot concordance group to the reals. Prima facie, these concordance invariants have the potential to provide independent bounds on the genus and number of double points for immersed surfaces with boundary a given knot.