Dehn filling and the Thurston norm

TL;DR

This paper studies how the Thurston norm behaves under Dehn filling of 3-manifolds with boundary, identifying finite exceptional slopes and generalizing previous results to broader classes of manifolds.

Contribution

It extends the understanding of Thurston norm behavior under Dehn filling to a wider class of 3-manifolds and addresses a specific question about knot genus after twisting.

Findings

Finite sets of slopes where Thurston norm behavior is unpredictable.

For all but finitely many slopes, Thurston norm plus winding norm equals original Thurston norm.

Application to knot theory answering a question about Seifert genus after twisting.

Abstract

For a compact, orientable, irreducible 3-manifold with toroidal boundary that is not the product of a torus and an interval or a cable space, each boundary torus has a finite set of slopes such that, if avoided, the Thurston norm of a Dehn filling behaves predictably. More precisely, for all but finitely many slopes, the Thurston norm of a class in the second homology of the filled manifold plus the so-called winding norm of the class will be equal to the Thurston norm of the corresponding class in the second homology of the unfilled manifold. This generalizes a result of Sela and is used to answer a question of Baker-Motegi concerning the Seifert genus of knots obtained by twisting a given initial knot along an unknot which links it.