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Title: Chern–Simons functional, singular instantons, and the four-dimensional clasp number
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.  more » « less
Award ID(s):
1952762 1952805
NSF-PAR ID:
10416221
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Journal of the European Mathematical Society
ISSN:
1435-9855
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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