Bordered invariants of pairs of sutured manifolds with torus boundary

TL;DR

This paper develops a framework to extend invariants of sutured manifolds to pairs differing by a basic slice attachment along a torus boundary, with applications to bordered-sutured Floer homology.

Contribution

It introduces a general framework for extending sutured manifold invariants to pairs differing by basic slice attachments, applicable to various invariants including Floer homology.

Findings

Framework recovers bordered-sutured Floer homology for specific manifold pairs.

Applicable to other invariants of sutured manifolds beyond Floer homology.

Broadens understanding of how sutured invariants behave under boundary modifications.

Abstract

We establish a framework for extending invariants of sutured manifolds to invariants of pairs of sutured manifolds who differ by attaching a basic slice along a torus boundary component. In the particular case of (bordered-)sutured Floer homology we show that this recovers the bordered-sutured Floer homology of a corresponding bordered-sutured three-manifold - though the framework is broad enough that it should be applicable in the setting of other invariants of sutured manifolds.