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F. Ye 11-18-2021

Title:  A large surgery formula for instanton Floer homology.
Abstract: For a knot K in the 3-sphere, Ozsváth-Szabó and Rasmussen introduced a large surgery formula which computes the Heegaard Floer homology of m-surgery on K for any large integer m, in terms of bent complexes defined using the knot Floer complex of K. In this talk, I’ll introduce an analogous formula for instanton Floer homology. More precisely, I construct two differentials on the instanton knot homology of K and use them to compute the framed instanton homology of m-surgery for any large integer m. As an application, I show that if the coefficients of the Alexander polynomial of K are not in {-1,0,1}, then there exists an irreducible representation from the fundamental group of S^3_r(K) to SU(2) for all but finitely many rational numbers r. In particular, all hyperbolic alternating knots satisfy this condition. Also by this large surgery formula, I show instanton and Heegaard knot Floer homology agree for any Berge knot, and that the framed instanton homology of S^3_r(K) agrees with the Heegaard Floer homology for any genus-one alternating knot K. This is a joint work with Zhenkun Li.
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