Bordered Floer homology and incompressible surfaces

TL;DR

This paper demonstrates that bordered Floer homology can detect incompressible surfaces and tangles, and extends computational algorithms to make these detections practical and combinatorial.

Contribution

It introduces new detection results for incompressible surfaces and tangles using bordered Floer homology, and extends algorithms for practical computation.

Findings

Bordered Floer homology detects incompressible surfaces.

Bordered-sutured Floer homology detects boundary parallel tangles.

Extended algorithms enable practical, combinatorial computations.

Abstract

We show that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Lambda so that the tensor product of CFD(Y) and Lambda is Hom-orthogonal to CFD(Y) if and only if the boundary of Y admits an essential compressing disk. In the process, we sharpen a nonvanishing result of Ni's. We also extend Lipshitz-Ozsv\'ath-Thurston's "factoring" algorithm for computing HF-hat to compute bordered-sutured Floer homology, to make both results on detecting incompressibility practical. In particular, this makes Zarev's tangle invariant manifestly combinatorial.