Not too long ago, in one of the final semesters of my undergraduate studies, I was programming in Topology. As a math enthusiast and programmer, I was working hard to complete a B.A. in Mathematics and a B.S. in Computer Science, and at the time, I was taking a few math upper-division electives that peeked my interest - the most interesting being Topology, of course. During that time, we were assigned two projects where we could do anything we wanted involving Topology up to what we had learned, and to my great creativity, I wrote two programs that solved (what could be considered) arduous tasks in the course.
- 2D-Mainfold Glueing (see below)
- calculate three qualities that form a complete invariant
- determine the homogeneity class
- Bracket/Jones Polynomial (see below)
- parameterize and digitize a knot in space
- calculate the bracket and Jones polynomial of that knot
Coding projects for Topology
This program, with an input polygon schema, can calculate the three qualities that form a complete invariant of 2-dimensional manifolds: the Euler characteristic, the orientability and, the genus of the resulting surface. The homogeneity class (the homogenous space formed) of the presented polygon schema is calculated with the three qualities above.
Using the "Walker parameterization" as the input of a knot (described in the attached paper), this program calculates the bracket polynomial and Jones polynomial of that knot.