Math Related Drawings

[Picture of a knotted surface] My primary research area is geometric topology. Lately I have been looking at knotted surfaces in 4-dimensional space. Here is a picture of a knotted surface. This is constructed from Fox's example 10 in the article A Quick Trip Through Knot Theory. You can make the image bigger by clicking on it. A similar picture appears on the cover of my book Knotted Surfaces and Their Diagrams with Masahico Saito

The main idea of the geometric topology of knotted surfaces is that surfaces in 4-space when they are projected to 3-space have self-intersection. To learn more read the book Knotted Surfaces and Their Diagrams Sometimes when you take an intersecting surface in 3-space, it won't lift to 4-space.

Boy Surface (illustrated here with a disk cut away) gives an immersion of a projective plane that has one triple point. This surface does not lift to an embedded surface in 4-space. The double cover of it is the surface that does not lift to an embedding in 4-space. This is illustrated next.

An immersed sphere in 3-space that doesn't lift to four dimensions is illustrated here. See if you can see that this represents a sphere by cutting and regluing the pieces along the double curves.

Koschorke's example is a surface formed from the connected sum of 3 Mobius bands that has one triple point. It has a double cover that does lift to 4-space even though it does lift to 4-space.

A knotted surface in 4-dimensional space is illustrated on the page The Seifert Algorithm.

Some Art


Etudes

Three peices that I did in an effort to understand the cartesean product of a pair of triangles. Prints of these can be ordered from me.

Etude3a. Etude4a. Etude5.

Seminar sketches

Boy's surface in movie form and the 3-sphere covering the quaternions.

Boy's Surface. quaternions.

trefoil. The handle structure in the trefoil complement (rough sketch)


Studies in Binomial and Multinomial

These are some completed things. There are some more in the works that help illustrate Pascal's recursion, and the multinomial recursion from a geomatric point of view.

Pascal cube. Observe that the 1,4,6,4,1 is represented by vertex,tetrahedon, octohedron, terahedron, vertex.

Multinomial. These are the structures for the trinomial theorem.

trinomial. A short improv on the same theme.



Don't forget to write.

J. Scott Carter
Professor of Mathematics
Department of Mathematics and Statistics
ILB 325
University of South Alabama
Mobile, AL 36688-0002

(251)-460-6264 /(251)-460-7969 FAX
click here for e-mail.

The views and opinions expressed in these web page(s) are strictly those of the author. The contents of these page(s) have not been reviewed or approved by the University of South Alabama. I am not responsible for content in linked material. Come to think of it, I am not repsonsible for much!