Guts in Sutured Decompositions and the Thurston Norm

TL;DR

This paper introduces a new invariant called guts for second homology classes in certain 3-manifolds, demonstrating its invariance properties and applications to knot complements and Kakimizu complexes.

Contribution

It constructs the guts invariant, proves its invariance under specific conditions, and relates it to sutured decompositions and other topological invariants.

Findings

Guts invariant is well-defined for second homology classes in irreducible 3-manifolds with toral boundary.

Guts of classes in the same Thurston cone are invariant under a natural condition.

The dimension of maximal simplices in Kakimizu Complexes is an invariant.

Abstract

We construct an invariant called guts for second homology classes in irreducible 3-manifolds with toral boundary and non-degenerate Thurston norm. We prove that the guts of second homology classes in each Thurston cone are invariant under a natural condition. We show that the guts of different homology classes are related by sutured decompositions. As an application, an invariant of knot complements is given and is computed in a few interesting cases. Besides, we show that the dimension of a maximal simplex in a Kakimizu Complex is an invariant.