On a Heegaard Floer theory for tangles

Abstract

The purpose of this thesis is to define a "local" version of Ozsv\'{a}th and Szab\'{o}'s Heegaard Floer homology $\operatorname{\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology $\operatorname{\widehat{HFT}}$ for tangles in the closed 3-ball. After studying basic properties of $\operatorname{\widehat{HFT}}$ and its decategorified tangle invariant $\nabla_T^s$, we prove a glueing theorem in terms of Zarev's bordered sutured Floer homology, which endows $\operatorname{\widehat{HFT}}$ with an additional glueing structure. For 4-ended tangles, we repackage this glueing structure into certain curved complexes $\operatorname{CFT}^\partial$, which we call peculiar modules. This allows us to easily recover oriented and unoriented skein relations for $\operatorname{\widehat{HFL}}$. Our peculiar modules enjoy some symmetry properties, which support a conjecture about $\delta$-graded mutation invariance of $\operatorname{\widehat{HFL}}$. In fact, we show that any two links related by mutation about a $(2,-3)$-pretzel tangle have the same $\delta$-graded link Floer homology. In the last part of this thesis, we explore the relationship between peculiar modules and twisted complexes in the fully wrapped Fukaya category of the 4-punctured sphere. This thesis is accompanied by two Mathematica packages. The first is a tool for computing the generators of $\operatorname{\widehat{HFT}}$ and its decategorified tangle invariant $\nabla_T^s$. The second allows us to compute Zarev's bordered sutured Floer invariants of any bordered sutured manifold using nice diagrams.