Lissajous Curves as A'Campo Divides, Torus Knots and Their Fiber Surfaces

Abstract

We present a new form of the fiber surface $F$ for links arising from isolated complex plane curve singularities, in particular, the torus knot $T(p, q)$, where $F$ is a smoothing of a long thin band which has as many clasp-singularities as the unknotting number of $T(p, q)$. As an application, we give a visual proof that the Lissajous curve $(\cos{p\theta}, \cos{q\theta})$ regarded as a divide corresponds to the torus link $T(p, q)$.