On a Heegaard Floer theory for tangles

Abstract

The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s Heegaard Floer homology HFL^ for links in the 3-sphere, i.e. a Heegaard Floer homology HFT^ for tangles in the 3-ball. The decategorification of HFL^ is the classical Alexander polynomial for links; likewise, the decategorification of HFT^ gives a local version ∇ˢ of the Alexander polynomial. In the first chapter of this thesis, we give a purely combinatorial definition of this polynomial invariant ∇ˢ via Kauffman states and Alexander codes and investigate some of its properties. As an application, we show that the multivariate Alexander polynomial is mutation invariant. In the second chapter, we define HFT^ in two slightly different, but equivalent ways: One is via Juhász’s sutured Floer homology, the other by imitating the construction of HFL^. We then state a glueing theorem in terms of Zarev’s bordered sutured Floer homology, which endows HFT^ with additional structure. As an application, we show that any two links related by mutation about a (2,−3)-pretzel tangle have the same δ-graded link Floer homology. This result relies on a computer calculation. In the third and last chapter, we specialise to 4-ended tangles. In this case, we give a reformulation of HFT^ with a glueing structure in terms of (what we call) peculiar modules. Together with a glueing theorem, we can easily recover oriented and unoriented skein relations for HFL^. Our peculiar modules also enjoy some symmetry relations, which support a conjecture about δ-graded mutation invariance of HFL^. However, stronger symmetries would be needed to actually prove this conjecture. Finally, we explore the relationship between peculiar modules and twisted complexes in the wrapped Fukaya category of the 4-punctured sphere. There are four appendices, some of which might be of independent interest: In the first appendix, we describe a general construction of dg categories which unifies all algebraic structures used in this thesis, in particular type A and type D modules from bordered theory. In the second appendix, we prove a generalised version of Kauffman’s clock theorem, which plays a major role for our decategorified invariants. The last two appendices are manuals for two Mathematica programs. The first is a tool for computing the generators of HFT^ and the decategorified tangle invariant ∇ˢ. The second allows us to compute bordered sutured Floer homology using nice diagrams.