Efficient construction of homological Seifert surfaces

Abstract

Let $\Omega$ be a bounded domain of $\mathbb{R}^3$ whose closure $\overline{\Omega}$ is polyhedral, and let $\mathcal{T}$ be a triangulation of $\overline{\Omega}$. Assuming that the boundary of $\Omega$ is sufficiently regular, we provide an explicit formula for the computation of homological Seifert surfaces of any $1$-boundary $\gamma$ of $\mathcal{T}$; namely, $2$-chains of $\mathcal{T}$ whose boundary is $\gamma$. It is based on the existence of special spanning trees of the complete dual graph of $\mathcal{T}$, and on the computation of certain linking numbers associated with those spanning trees. If the triangulation $\mathcal{T}$ is fine, the explicit formula is too expensive to be used directly. For this reason, making also use of a simple elimination procedure, we devise a fast algorithm for the computation of homological Seifert surfaces. Some numerical experiments illustrate the efficiency of this algorithm.