Exotic disks and singular instanton Floer homology

Abstract

We show that singular instanton Floer homology with the Chern--Simons filtration can be used to produce exotic pairs of slice disks. We moreover construct a strongly invertible $\mathbb{Z}$-slice knot for which any symmetric pair of $\mathbb{Z}$-disks are exotic, and remain exotic after stabilizing by $n\smash{\mathbb{CP}}^2$ or $n\smash{\overline{\mathbb{CP}}}^2$ (or by standard $n\smash{\mathbb{RP}}^2$ or $-n\smash{\mathbb{RP}}^2$) for any $n$. Our methods apply more generally to stabilization by any simply connected definite manifold, or by any number of exotic embedded projective planes of the same sign. We also provide an example of a strongly invertible knot which is $\mathbb{Z}$-slice and equivariantly slice, but not equivariantly $\mathbb{Z}$-slice. Along the way, we partially compute various symmetry actions on the singular instanton Floer complexes of two-bridge knots via an explicit analysis of their traceless $\mathit{SU}(2)$-character varieties.