# Saul Schleimer and Henry Segerman

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 a research fellow in the Department of Mathematics and Statistics at the University of Melbourne. His mathematical research is in 3-dimensional geometry and topology, and concepts from those areas often appear in his work. Other artistic interests involve procedural generation, self reference, ambigrams and puzzles.

Many of our recent projects have involved objects native to the 3-sphere, realized in Euclidean space via stereographic projection. This project continues the theme: we realize Seifert surfaces for the (4,3), (5,3) and (5,4) torus knots, and for the (3,3) torus link, as leaves of the associated Milnor fibration.

As elegantly discussed in Ghys' 2006 ICM plenary talk, the natural
parameterization of the Seifert surface for the trefoil knot uses
Eisenstein series of lattices in the plane. This was generalized
by Milnor to all (p,q) torus knots; he replaces Eisenstein series
by certain fractional automorphic forms. Tsanov reduces the
construction of these forms to finding an analytic description of
the universal covering of the orbifold S^2(p,q,infinity) by the
hyperbolic plane. Mainly following Lehner, we find a Fourier
series for the covering map. Combining these ideas, we obtain a
map from a hyperbolic triangle T_H, with angles pi/p, pi/q, and 0,
to a domain T_S in S^3; rigid symmetries of T_S in S^3 generate
the Seifert surface.

Using Schwarz-Christoffel theory we uniformize T_H by a Euclidean
triangle T_E having angles pi/p, pi/q and pi(1-1/p-1/q). In this
way we transfer decorations on T_E to the Seifert surface; for
these sculptures we use a subdivision of T_E into 15 congruent
triangles.