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.
KLB was partially supported by a grant from the Simons Foundation (#209184 to Kenneth L. Baker). SAT was supported by a grant from the Natural Science Division of Colby College.
"Dehn filling and the Thurston norm." J. Differential Geom. 112 (3) 391 - 409, July 2019. https://doi.org/10.4310/jdg/1563242469