A Seifert algorithm for integral homology spheres

Abstract

From classical knot theory we know that every knot in $S^3$ is the boundary of an oriented, embedded surface. A standard demonstration of this fact achieved by elementary technique comes from taking a regular projection of any knot and employing Seifert's constructive algorithm. In this note we give a natural generalization of Seifert's algorithm to any closed integral homology 3-sphere. The starting point of our algorithm is presenting the handle structure of a Heegaard splitting of a given integral homology sphere as a planar diagram on the boundary of a $3$-ball. (For a well known example of such a planar presentation, see the Poincar\'e homology sphere planar presentation in {\em Knots and Links} by D. Rolfsen \cite{Rolfsen}.) An oriented link can then be represented by the regular projection of an oriented $k$-strand tangle. From there we give a natural way to find a ``Seifert circle" and associated half-twisted bands.