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

  • Aliakbar Daemi

    Washington University in St. Louis, USA
  • Christopher Scaduto

    University of Miami, Coral Gables, USA
Chern–Simons functional, singular instantons, and the four-dimensional clasp number cover

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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 quasialternating 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 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.

Cite this article

Aliakbar Daemi, Christopher Scaduto, Chern–Simons functional, singular instantons, and the four-dimensional clasp number. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1320