Seifert surface for (3,3) torus link

This is joint work with Saul Schleimer.

A torus link is a link that can be drawn on a torus. A Seifert surface spans its link, somewhat like a soap-film clinging to its supporting wire-frame. The surface acts as a bridge between the 1-dimensional link and the 3-dimensional space it lives in.

The torus links and their Seifert surfaces live most naturally in the 3-sphere, a higher dimensional version of the more familiar sphere. We transfer our sculptures to Euclidean 3-space using stereographic projection. The Seifert surface is cut out of the 3-sphere by the Milnor fibers of the corresponding algebraic singularity. We parametrize the Milnor fiber, following the work of Tsanov, via fractional automorphic forms. These give a map from SL(2,R), the canonical geometry of the torus link complement, to the 3-sphere.

The patterns on each Seifert surface arise from two applications of the Schwarz-Christoffel theory of complex analysis, turning a Euclidean triangle into a hyperbolic one. We used our maker's mark for the pattern.