2019 Icerm Illustrating Mathematics

# François Guéritaud, Saul Schleimer, and Henry Segerman

## Artists

## Statement

François Guéritaud is a CNRS researcher at the Laboratoire Paul-Painlevé, Lille. He likes to mediate an interest in geometric topology and combinatorics through arts and crafts. 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 figure-eight knot, the most common knot after the trefoil, is widely used in sailing and rock-climbing. The sculpture here contains the complement of the figure-eight knot - the part of space outside of the knot. The core curve is parametrised in the three-sphere then stereographically projected to three-space. Let $A(t) = \varepsilon \sin 4t$. Then the parametrization is given by
$w(t) = (1 - A(t)^2) (\lambda \sin t - (1 - \lambda) \sin 3t)/(1 + A(t)^2)$
$x(t) = (1 - A(t)^2) (\lambda \cos t + (1 - \lambda) \cos 3t)/(1 + A(t)^2)$
$y(t) = (1 - A(t)^2) (2\sqrt{\lambda - \lambda^2} \sin 2t)/(1 + A(t)^2)$
$z(t) = 2 A(t)/(1 + A(t)^2)$
We chose the values $(0.25,0.16)$ for the parameters $(\lambda, \varepsilon)$.