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

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 Frøyshov 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.

68 pages; 6 figures; section on knot ideals removed and reworked into 2209.05400; published in JEMS

References in corpus (6)

• Knot homology groups from instantons
• Gauge theory and Rasmussen's invariant
• The slicing number of a knot
• On a Nonorientable Analogue of the Milnor Conjecture
• Concordance invariants from the $E(-1)$ spectral sequence on Khovanov homology
• Instantons, Bar-Natan homology, and some concordance invariants of knots

Cited by in corpus (8)

• Instantons, special cycles, and knot concordance
• From zero surgeries to candidates for exotic definite four-manifolds
• Involutions, knots, and Floer K-theory
• On the nonorientable four-ball genus of torus knots
• Instanton knot invariants with rational holonomy parameters and an application for torus knot groups
• Slice genus, $T$-genus and $4$-dimensional clasp number
• Heegaard Floer homology and plane curves with non-cuspidal singularities
• Cables of the figure-eight knot via real Frøyshov invariants