Thurston norm

Definition

Let $M$ be a compact orientable 3-manifold. For a compact surface $S$ with connected components $S_1, \ldots, S_k$, set $$\chi_-(S) = \sum_{i=1}^{k} \max(0, -\chi(S_i)),$$ where $\chi$ is the Euler characteristic. For a class $a \in H_2(M, \partial M; \Z)$ in homology, the Thurston norm of $a$ is $$x(a) = \min \set{\chi_-(S) : S \text{ a properly embedded oriented surface in } M \text{ with } [S] = a}.$$ The function $x$ satisfies $x(n a) = n\, x(a)$ for integers $n \ge 0$ and $x(a + b) \le x(a) + x(b)$, and it extends uniquely to a continuous seminorm on $H_2(M, \partial M; \R)$, also called the Thurston norm.