Saul Schleimer and Henry Segerman

Reader; Associate Professor of Mathematics
Mathematics Institute, University of Warwick; Department of Mathematics, Oklahoma State University
Coventry, United Kingdom; Stillwater, Oklahoma, USA
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 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.
$(3,3)$ Seifert Surface
16 x 14 x 15 cm
PA 2200 Nylon Plastic, Selective-Laser-Sintered; Rit Dye
The Seifert surface of a torus knot (or link) can be realized as a \emph{Milnor fiber}; we use fractional automorphic forms to obtain a parametrisation. The fiber is then thickened using the so-called \emph{elliptic flow}. This flow comes from the action of $\SO(2)$ subgroup on $\widetilde{\PSL(2,\RR)}$, pushed down to $S^3$. 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.
$(3,3)$ Seifert Surface with Fibers
21 x 19 x 21 cm
PA 2200 Nylon Plastic, Selective-Laser-Sintered; Rit Dye
When $p = q$, the flow lines of the elliptic flow are all fibers of the \emph{Hopf fibration}: the fibration of $S^3$ coming from intersecting $S^3$ with all complex lines in $\CC^2$. 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 off.