2019 Icerm Illustrating Mathematics

# Saul Schleimer and Henry Segerman

## Artists

## Statement

Saul Schleimer is a geometric topologist, working at the University of Warwick. His other interests include combinatorial group theory and computation. He is especially interested in the interplay between these fields and additionally in the visualization of ideas from these fields. Henry Segerman is an Associate Professor in the Department of Mathematics at Oklahoma State University. His interests include geometry and topology, 3D printing, virtual reality, and spherical video.

## Artworks

The Seifert surface of a torus knot (or link) can be realized as a Milnor fiber; we use fractional automorphic forms to obtain a parametrization. The fiber is then thickened using the so-called elliptic flow. This flow comes from the action of SO(2) subgroup on PSL(2,R), pushed down to the three-sphere. The flow is transverse to the Seifert surface away from the boundary. As a point x approaches the boundary, the flow line through x topples. Finally, the boundary is a union of flow lines.
Since p = q = 3, a highly symmetric pattern is possible for the fundamental domain of the tiling of the surface.

When p = q, the flow lines of the elliptic flow are all fibers of the Hopf fibration: the fibration of the three-sphere coming from intersecting it with all complex lines in complex two-space. In this piece, we add to the (3,3) Seifert surface all flow lines through the small hexagons of the tiling of the Seifert surface. Flow lines going too close to infinity are cut away.