Chern-Simons functional, singular instantons, and the four-dimensional clasp number

TL;DR

This paper explores the relationship between the four-dimensional clasp number and slice genus using a knot invariant from equivariant singular instanton theory, providing new insights into knot concordance and related conjectures.

Contribution

It introduces a new invariant derived from singular instanton theory related to the Chern--Simons functional, addressing longstanding questions in knot theory and 4-manifold topology.

Findings

The invariant can distinguish differences between clasp number and slice genus.

Computed the invariant for quasi-alternating and torus knots, relating it to the knot signature.

Provided evidence supporting an extension of the slice-ribbon conjecture to torus knots.

Abstract

Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern--Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Fr{\o}yshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless $SU(2)$ representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.