![[Picture of a knotted surface]](example10.jpg)
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 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.
Waving windows to the 4th dimension.
I had the same dream a week later.
My dream under the intense scrutiny of psychoanalyis.
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
trefoil. The handle structure in the trefoil complement (rough sketch)
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! |
