$\operatorname{SL}(2,\mathbb{C})$ Floer Homology for surgeries on some knots

Ikshu Neithalath

Published 2020-02-10Version 1

We establish a relationship between the sheaf-theoretic $\operatorname{SL}(2,\mathbb{C})$ Floer cohomology $\mathit{HP}(Y)$, as defined by Abouzaid and Manolescu, for $Y$ a surgery on a small knot in $S^3$, and the $\operatorname{SL}(2,\mathbb{C})$ Casson invariant, as defined by Curtis. We use this to compute $\mathit{HP}$ for surgeries on the trefoil and the figure-eight knots. We also compute $\mathit{HP}$ for surgeries on two non-small knots, the granny and square knots.