Geometry and Topology Seminar
In Heegaard Floer homology, Oszváth-Szabó and Rasmussen introduced a large surgery formula computing HF^\hat(S^3_m(K)) for any knot K and large integer m by bent complexes from CFK^-(K). In this talk, I'll introduce a similar formula for instanton Floer homology. More precisely, I construct two differentials on the instanton knot homology KHI(K) and use them to compute the framed instanton homology I^#(S^3_m(K)) for any large integer m. As an application, I show that if the coefficients of the Alexander polynomial of K are not ±1, then there exists an irreducible SU(2) representation of the fundamental group of S^3_r(K)) for all but finitely many rational r. In particular, all hyperbolic alternating knots satisfy this condition. Also by this large surgery formula, I show KHI(K)=HFK^\hat(K) for any Berge knot and I^#(S^3_r(K))=HF^\hat(S^3_r(K)) for any genus-one alternating knot.