Framed instanton homology and concordance

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