Decomposing sutured monopole and Instanton Floer homologies

TL;DR

This paper advances sutured monopole and instanton Floer theories by establishing grading properties, algorithms, decompositions, and norm detection, extending previous results to links and providing new computational tools.

Contribution

It introduces a grading shifting property, algorithms for Floer homology computation, canonical decompositions, and extends knot Floer homology constructions to links.

Findings

Provides an algorithm for Floer homology of sutured handle-bodies

Establishes a Thurston-norm detection result for knot Floer homologies

Proves that Floer homologies bound the depth of sutured manifolds

Abstract

In this paper, we generalize the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juh\'asz for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for monopole and instanton knot Floer homologies, which were introduced by Kronheimer and Mrowka. The same result was originally proved by Ozsv\'ath and Szab\'o for link Floer homology. Last, we generalize the construction of minus versions of monopole and instanton knot Floer homology, which was initially done for knots by the second author, to the case of links. Along with the construction of polytopes, we also proved that, for a balanced sutured manifold with vanishing second homology, the rank of the sutured monopole or instanton Floer homology bounds the depth of the balanced sutured manifold. As a corollary, we obtain an independent proof that monopole and instanton knot Floer homologies, as mentioned above, both detect fibred knots in $S^3$. This result was originally achieved by Kronheimer and Mrowka.