  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.
