A Curve Complex and Incompressible Surfaces in $S\times \mathbb{R}$

Various curve complexes with vertices representing multicurves on a surface $S$ have been defined, for example [3], [4] and [8]. The homology curve complex $\mathcal{HC}(S,\alpha)$ defined in [7] is one such complex, with vertices corresponding to multicurves in a nontrivial integral homology class $\alpha$. Given two multicurves $m_1$ and $m_2$ corresponding to vertices in $\mathcal{HC}(S,\alpha)$, it was shown in [8] that a path in $\mathcal{HC}(S,\alpha)$ connecting these vertices represents a surface in $S\times \mathbb{R}$, and a simple algorithm for constructing minimal genus surfaces of this type was obtained. In this paper, a Morse theoretic argument will be used to prove that all embedded orientable incompressible surfaces in $S\times \mathbb{R}$ with boundary curves homotopic to $m_{2}-m_1$ are homotopic to a surface constructed in this way. This is used to relate distance between two vertices in $\mathcal{HC}(S,\alpha)$ to the Seifert genus of the corresponding link in $S\times \mathbb{R}$.