This research project is continuation of the project I have worked on during the Fall
semester 2002 under Dr. David T. Gay’s supervision. So far, we have studied different
functions acting on topological surfaces, mainly, smooth, invertible, functions from a
surface to itself. The research has been focused on a specific class of functions on
surfaces, namely, compositions of twists along simple closed curves. The braid relations,
Lantern relation, and the Chain relation, which have been proven in the course of the
project, are good example of working with such functions.
The research on this project will be primarily focused on finding boundary-interior
relations for different surfaces as well as proving these relations by means of diagrams
similar to the ones already used. The main interest will be finding twelve curves Ci, i = 1,
…, 12 in a torus with nine boundary components such that the composition of twists
along C1, …, C12 equals one right twist along each boundary of the nine punctured torus.
To insure progress, I will meet with Dr. Gay on a weekly basis. During each meeting,
Dr. Gay will check my work and suggest any alternative ways, if any, that could simplify
the work. At the conclusion of my research project, I intend to write a full report suitable