## Math Related Drawings

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

#### Works from 2010

improvisation #1 playing with a weird projection of the triangle times triangle

improvisation #2 again.

improvisation #3 and again.

improvisation #4 and once more.

tiles 1: I was doing a calculation on a nine-element strict 2-quandle that I encoded here.

tiles 2: Same calculation, a little improvisation.

tiles 3: Same calculation, a little more improvisation.

tiles 4: Same calculation, a little more improvisation.

tiles 5: Same calculation, a little more improvisation.

#### Works from 2009

Velvet Elvis # 1. A late night improvisation on the double point set of the sphere eversion near the quadruple point.

One Quarter An improvisation on the 4 copies of the triangle times the triangle that fill a hypercube.

Cirque de Soleil series: # 1, # 2, # 3, # 4, #5, and the contortionists: #6.

Triangle 3 v2 c The ocean of fish in the net is how one of the above should look. The computer needs to free its mind, No?

Triangle 3 v2 b Corrected version.

Triangle 3 v2 d Corrected version.

Triangle # 4 A work developed for an up coming math art conference.

An Ocean in New Mexico An improvisation on the same data. Appropriate for a Doctor's office, no?

My computer is taking LSD. File transfer problems for Triangle 3 version 2a .

My computer is still tripping. File transfer problems for Triangle 3 version 2b .

Triangle 3 v2 d1 Maybe the computer is starting to come down.

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.

#### Seminar sketches

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