Dehn filling and the Thurston norm (Q2315241)

Let \(N\) be a compact, orientable 3-manifold whose boundary consists of tori, and let \(T\) be a particular boundary component. Let \(N_T(b)\) be the manifold obtained by Dehn filling \(N\) along a boundary slope \(b\) in \(T\). This paper considers the behavior of the Thurston norm under Dehn filling. If \(z\in H_2(N,\partial N-T)\) is represented by an embedded surface \(S\) in \(N\) disjoint from \(T\), then the corresponding class \(\hat z \in H_2(N_T(b),\partial N_T(b))\), defined by inclusion, is also represented by \(S\), and then \(x(z)\geq x(\hat z)\), where \(x(-)\) denotes the Thurston norm in the corresponding homology group (roughly speaking, the Thurston norm of a homology class is the minimum of minus the Euler characteristic of the surfaces representing the given class). \textit{D. Gabai} [J. Differ. Geom. 26, 461--478 (1987; Zbl 0627.57012)] showed that for a fixed class \(z\in H_2(N,\partial N-T)\), \(x(z)=x(\hat z)\) for all except at most one slope \(b\) in \(T\). \textit{Z. Sela} [Isr. J. Math. 69, No. 3, 371--378 (1990; Zbl 0714.57009)] extended this, showing that for every class \(z\), \(x(z)=x(\hat z)\), for all Dehn fillings except for a finite number of slopes \(b\) in \(T\). In the paper under review, all classes in \(H_2(N,\partial N)\) are considered. If \(\hat z \in H_2(N_T(b),\partial N_T(b))\) is represented by a surface \(\hat S\) transverse to \(K_b\), where \(K_b\) is the core of the filled solid torus, then the class of \(S = \hat S \cap N\) is represented by a class \(z\in H_2(N,\partial N)\). The absolute value of the algebraic intersection number between \(K_b\) and \(\hat z\) is called the winding number of \(K_b\) along \(\hat z\) and denoted by \(\mathrm{wind}_{K_b}(\hat z)\). It follows that \(x(z)\geq x(\hat z)+\mathrm{wind}_{K_b}(\hat z)\). If this inequality is in fact not an equality, then \(b\) is called a norm reducing slope. The main result of the paper shows that if \(N\) is not \(T\times I\) nor a cable space, then for each torus boundary component \(T\) of \(\partial N\), there is a finite set of slopes \(\mathcal{R}=\mathcal{R}(N,T)\) in \(T\) such that if \(b\notin \mathcal{R}\) then \(b\) is not norm reducing. A bound for the size of the set \(\mathcal{R}\) is also given. Let \(\{ K_n \}\) be a family of knots in \(S^3\) obtained by performing \(-1/n\)-Dehn surgery along an unknot \(c\) that links a given knot \(K_0\), and let \(g(K)\) denote the genus of \(K\). As an application, it is shown that if \(lk(K_0,c)\not= 0\), then \(g(K_n)\rightarrow \infty\) as \(n\rightarrow \infty\), unless \(c\) is a meridian of \(K_0\). This improves a previous result of \textit{K. L. Baker} and \textit{K. Motegi} [Commun. Anal. Geom. 27, No. 4, 743--790 (2019; Zbl 1432.57002)].