Lissajous curves as A'Campo divides, torus knots and their fiber surfaces (Q1812501)

N. A'Campo associated a link in \(S^3\) to a divide, a relative, generic immersion of a 1-manifold in a unit disk in \(\mathbb{R}^2\), and M. Hirasawa gave an algorithm to draw the link diagram of a divide. In this paper the authors give a visual proof that the link associated to the Lissajous curve (\(\cos p\theta\), \(\cos q\theta\)), regarded as a divide, is given by the torus link \(T(p,q)\). The approach to show this is to construct the link diagram and the fiber surface \(F\) for a Lissajous divide explicitly, where \(F\) is obtained as a smoothing of a long thin band having as many clasp-singularities as the unknoting number of \(T(p,q)\). The authors claim that such a construction of fiber surfaces is also available for any link arising from isolated complex curve singularities.