On simple loops on a solid torus of general genus

Abstract:

Let $l$ be a simple loop on the boundary of a solid torus $T$ of genus $g$. Let ${\mathbf {m}}$ be a complete system of oriented meridian disks of $T$. Let $W(l,{\mathbf {m}})$ be a word obtained by reading the intersections $l \cap {\mathbf {m}}$ along $l$. We shall give a natural method for realizing the cyclic reductions of $W(l,{\mathbf {m}})$ geometrically. This yields a simple proof of Whitehead-Zieschang’s theorem related to the minimality of the intersections of two systems $\{ {l_1}, \ldots ,{l_n}\}$ and ${\mathbf {m}}$.