On computation of morphism spaces and a direct limit of the bordered Floer homology of knot complements

TL;DR

This paper investigates the direct limit of bordered Floer homology for knot complements, focusing on morphism spaces and a new invariant that detects non-unstable chains, advancing understanding of knot invariants.

Contribution

It constructs a direct system of positively framed knot complements and introduces a direct limit invariant, linking morphism spaces to knot complement properties in bordered Floer theory.

Findings

Constructed a direct system of knot complements with increasing framing.

Derived type-DA morphisms from DD morphisms within the system.

Introduced a new invariant detecting non-unstable chains in type-D modules.

Abstract

In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism space between two type-DD modules, and derive type-DA morphisms from DD morphisms to derive the direct system maps. In addition, we introduce a direct limit invariant from the direct system which can detect non-unstable chains in the type-D module of a knot complement, if the type-D modules of the direct system are obtained by algorithm of Lipshitz, Ozsvath and Thurston.