SL(2,C) Floer Cohomology for Surgeries on Some Knots

Abstract

We establish a relationship between $\mathrm{HP}(\mathit{Y})$, the Abouzaid–Manolescu sheaf-theoretic $\mathrm{SL}(2,\mathbb{C})$ Floer cohomology for a surgery Y on a small knot in ${\mathit{S}}^3$ and Curtis’ $\mathrm{SL}(2,\mathbb{C})$ Casson invariant. We use this to compute $\mathrm{HP}$ for most surgeries on two-bridge knots. We also compute $\mathrm{HP}$ for surgeries on two nonsmall knots, the granny and square knots. We provide a partial calculation of the framed sheaf-theoretic $\mathrm{SL}(2,\mathbb{C})$ Floer cohomology ${\mathrm{HP}}_{\mathrm{\#}}(\mathit{Y})$ for surgeries on two-bridge knots and apply the data we obtain to show the nonexistence of a surgery exact triangle for ${\mathrm{HP}}_{\mathrm{\#}}$.