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.

Figures (6)

• Figure 1: The double twist knot $D_{m,n}$ has $2m$ and $2n$ half twists in the indicated boxes. The example shown has $m=n=2$, and is the two bridge knot $(15,4)$. The two indicated band moves induce a genus 1 cobordism from $D_{m,n}$ to the unknot.
• Figure 2: Above are generators of $C_\ast(D_{2,2};\Delta_\mathscr{S})$ along with the reducible $\zeta^0$. Bigradings are listed under each corresponding generator. Arrows are multiplication by $T^2-T^{-2}$ and represent $d$ except for the left-most arrow, which is $\delta_1$. The component $\delta_2$ is zero. While $v$ might be nonzero, there is no cycle that $v$ maps to $\zeta^3$. The local chain equivalence to $\widetilde{{\mathfrak C}}(t)$ with $t=\frac{9}{15}$ leaves only $\zeta^0\leftarrow \zeta^3$ in this diagram.
• Figure 3: This diagram shows generators for $C_\ast(K;\Delta_\mathscr{S})$ and the reducible $\zeta^0$ for $K=K_{51,16}$. As in Figure \ref{['fig:74complex']}, all arrows are multiplication by $\pm (T^2-T^{-2})$. The short (blue) arrows represent $d$ except for the left-most one, which is $\delta_1$. Again, $\delta_2=0$. The two L-shaped (green) arrows are multiplication by $\pm(T^2-T^{-2})$, representing components of $v$. The $v$-map might have other non-zero components, but the only ones that contribute to $\Gamma_{K}$ are the ones shown.
• Figure 4: Crossing changes realizing the unknotting numbers of the indicated knots.
• Figure 5: The knot $K'$ is obtained from $K$ by blowing up around the strands indicated, which introduces one full right-handed twist. The linking number $d$ is the number of upwards pointing strands minus the number of downwards pointing strands.

Theorems & Definitions (127)

• Theorem 1
• Corollary 1
• Theorem 2
• proof : Proof of Theorem \ref{['thm:clasp']}
• Theorem 3
• Theorem 4
• Remark 1
• Theorem 5
• Theorem 6
• Theorem 7