Scott Carter

Professor Emeritus, Professor
University of South Alabama, George Washington University
Osaka City University, Osaka, Japan

Inspired by on-going conversations with Nancy Hocking, I created this illustration of the 2-twist-spun trefoil. It is the most simple known knotted sphere in 4-dimensional space that projects into 3-dimensional space with exactly four triple points. Various diagrams have been created of this knotted sphere. It was first discovered by Ralph Fox, and later found to be one in an infinite family of knotted spheres by Christopher Zeeman. This drawing is also based on the descriptions that were given by Shin Satoh, Akiko Shima, Ayumu Inoue, and Kengo Kawamura. The technique of rendering is a modification of that given by Masahico Saito and myself as influenced by Dennis Roseman.

The 2-twist-spun-trefoil
The 2-twist-spun-trefoil
90 x 60 cm
digital print

This is a broken surface diagram of the 2-twist-spun trefoil that is created with a one point perspective and a movie presentation. Horizontal slices lie upon parallel hyperplanes with infinitesimal thickness. These slice the knotted sphere in classical knot diagrams. The perspective point of view causes a reflection from the viewers point of view. That the sphere is knotted can be seen since it is three colorable.