Seifert surface for (3,3) torus link

thingiverse

In collaboration with Saul Schleimer, we explore torus links on a torus, where Seifert surfaces span and bridge the 1-dimensional link to its 3-dimensional space habitat. The natural dwelling of these links and their surfaces is in the 3-sphere, a higher dimensional sphere version. To present our sculptures in Euclidean 3-space, we utilize stereographic projection. Seifert surfaces are formed by cutting the 3-sphere through Milnor fibers of associated algebraic singularities. We map these via fractional automorphic forms and SL(2,R) geometry, which represents the canonical geometry of torus link complements. The patterns on each surface come from two applications of Schwarz-Christoffel theory, transforming a Euclidean triangle into a hyperbolic one, marked by our unique signature.