Speaker:
Title:
Abstract: Every knot type in three-space may be realized as the boundary of a surface in three-space. This is useful because it is easier to distinguish two surfaces topologically than it is to distinguish knot types. In particular, it is easy to define a "linking matrix" associated to a surface in three-space. If two surfaces are topologically equivalent, then they will have congruent linking matrices. However, using surfaces to distinguish knots in this way is complicated by the fact that a given knot type occurs as the boundary of many topologically different surfaces.
Two knots are called S-equivalent if they cannot be distinguished by the linking matrices of the surfaces they bound. Recently, S. Naik and I have shown that S-equivalence is generated by a single combinatorial move on knot diagrams. This is not very useful computationally, but I believe it will be useful in understanding how S-equivalence is related to other approaches to the problem of classifying knots.
I will begin with a number of very elementary examples of surfaces and the knots that form their boundaries. I will make some use of scissors, paper, tape, and colored markers. I will describe the linking matrix and how to compute it. Then I will describe the difficulty encountered in trying to use these matrices to distinguish knots. By the end of the talk I hope to be able to at least state our recent result, and possibly explain an idea or two used in the proof.