  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.
