Framed instanton homology and concordance

We define two concordance invariants of knots using framed instanton homology. These invariants ${\nu }^{\mathrm{♯}}$ and ${\tau }^{\mathrm{♯}}$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $\tau$ invariant in Heegaard Floer homology. We use ${\nu }^{\mathrm{♯}}$ and ${\tau }^{\mathrm{♯}}$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyperbolic manifolds in the Hodgson–Weeks census. In another application, we determine when the cable of a knot is an instanton L-space knot. Finally, we discuss applications to the spectral sequence from odd Khovanov homology to the framed instanton homology of branched double covers, and to the behaviors of ${\tau }^{\mathrm{♯}}$ and $\tau$ under genus-2 mutation.