Divides with cusps, shadows, and transvergent diagrams

A link in $S^{3}$ is called a symmetric link if there is an orientation preserving involution of $S^{3}$ with a non-empty fixed point set that preserves the link. Any symmetric link can be depicted by a diagram with a symmetry axis lying on the plane of the diagram, called a transvergent diagram. Recently, Sugawara proved that any symmetric link can be represented by a divide with cusps, which is a generalization of A'Campo's divide that allows a finite number of cusps. In this paper, we introduce a generalization of A'Campo's divide in terms of Turaev's shadow, called a divide with gleams. By using divides with gleams, we provide an algorithm to obtain a divide with cusps that represents a symmetric link from its given transvergent diagram. Conversely, we also provide an algorithm to draw a transvergent diagram of the link of a given divide with cusps. Furthermore, we characterize freely $2$-periodic links through divides with gleams on a $2$-sphere.