A sheaf-theoretic SL(2,C) Floer homology for knots
Geometric Topology
2019-06-19 v2 Algebraic Geometry
Symplectic Geometry
Abstract
Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invariant is a sheaf-theoretic SL(2,C) analogue of the singular knot instanton homology of Kronheimer and Mrowka. We prove that for two-bridge and torus knots, the SL(2,C) invariant is determined by the l-degree of the -polynomial. However, this is not true in general, as can be shown by considering connected sums of knots.
Keywords
Cite
@article{arxiv.1811.07000,
title = {A sheaf-theoretic SL(2,C) Floer homology for knots},
author = {Laurent Côté and Ciprian Manolescu},
journal= {arXiv preprint arXiv:1811.07000},
year = {2019}
}
Comments
Several minor changes; final version, to appear in Proc. London Math. Soc