﻿ Visualization of Seifert Surfaces

Visualization of Seifert Surfaces

 This website is dedicated to the visualization of Seifert surfaces. The first question is obvious. What is a Seifert Surface? That was my first response when my colleague Arjeh Cohen asked me in May 2004 if I could make pictures of these. Since then, things got out of hand. Inspired and guided by Arjeh, I became totally intrigued by these strange and difficult surfaces and found it a challenge how to generate and display them. In contrast to many other mathematical objects, Seifert surfaces are very real-world and not abstract. Still their shapes are hard to comprehend, which makes them a good case for visualization.

 One result is a tool, called SeifertView. With this tool a variety of knots and links in many different styles, and with many display options can be produced. Knots and links are defined using the braid notation. I tried to design the user interface such that an occasional user can generate and view knots and surfaces, while also many options for tuning are available. SeifertView can be downloaded for free. We have described our work in a paper [1] for Visualization 2005. This paper provides a very short introduction to topology and knot theory, and the ideas behind the various methods used for generating Seifert surfaces. In the version of SeifertView available for download also some new methods are introduced. The most important of these is the incorporation of a physical model for smoothing, based on the work of Robert Scharein. More details on this can be found in [2]. Also, we have contributed images to the wonderful Mathematical Imagery website of the American Mathematical Society. Bathsheba Grossman, an American artist exploring the region between art and mathematics, has used SeifertView to produce wonderful 3D sculptures. George Hart has used SeifertView in one of his fascinating video essays for the Simons Foundation, as an example for 3D printing (see about 2:45). Finally, I want to thank everybody who has contributed to this work, and hope that you enjoy Seifert surfaces as much as I do! Jack van Wijk, September 2005
 [1] Jarke J. van Wijk, Arjeh M. Cohen. Visualization of the Genus of Knots. In:  C. Silva, E. Gröller, H. Rushmeier (eds.), Proceedings IEEE Visualization 2005, IEEE CS Press. [2] Jarke J. van Wijk, Arjeh M. Cohen. Visualization of Seifert Surfaces . IEEE Trans. on Visualization and Computer Graphics, vol. 12, no. 4, p. 485-496, 2006. .