My Research

Here's a brief description of some of the work I have done in mathematics that made it into publication.

Ph.D. Dissertation

I completed my dissertation in May 1977. Most of the ideas in that work came from or were inspired by my major professor, De Witt Sumners. He deserves a lot of credit for providing guidance to me while I was working on this.

Slice Knots which are not Ribbon Knots

A major unsolved problem in classical knot theory is to determine if every slice knot is a ribbon knot. In 1969, T. Yanagawa gave an example of a slice 2-knot which was not ribbon. I produced classes of examples in all higher dimensions of slice n-knots (n>1) which are not ribbon knots in my paper

Different Disk Knots with the same Exterior

In the early 1960's, Gluck showed that there can be at most two different n-sphere knots with the same exterior (n>1). Then in 1976, examples of inequivalent higher-dimensional knots with the same exterior were discovered by Capell-Shaneson and Gordon. The situation with knotted disk pairs is quite different as DeWitt Sumners and I showed in our two papers

In the first of these papers, we constructed infinitely many disk knot exteriors each one of which is the exterior of six different knotted disk pairs of the 5-disk in the 7-disk all having the same exterior. Through spinning, these examples gave similar examples in all higher dimensions. For knots of the 4-disk in the 6-disk, we obtained a similar result but could only produce exteriors corresponding to three different disk knots.

In the second article we refined the combinatorial group theory we used and were able to construct classes of inequivalent disk pairs of any given size all of which have the same exterior. But alas, we did not have enough examples of groups to find a single class of disk knots of infinite order having the same exterior. Steve Plotnick was able to find such a class of disk pairs using some PSL groups he had studied.

The idea for these two papers actually first appeared in an earlier joint paper with DeWitt.

Distinguishing Certain Ribbon Knots with the Jones Polynomial

In their 1988 paper, Ian Aitchison and Dan Silver showed (among other results) that there are exactly three S-equivalence classes of genus 2 fibered doubly slice knots. Dan and I developed results on the affect on the Jones polynomial when bands of a ribbon knot are twisted together or are passed through one another. Using this information on the Jones polynomial, we showed that each of the three S-equivalence classes of genus 2 fibered doubly slice knots in the 3-sphere can be represented by infinitely many distinct prime fibered doubly slice ribbon knots.

Helicoidal Surfaces with Constant Mean Curvature

A helicoidal surface is one which is invariant under some helicoidal motion in 3-space. Such surfaces which have constant mean curvature were described parametrically in 1982 by Do Carmo and Dajczer. I was the first person to generate computer images of these surfaces. Based on the images, I was able to make a couple of conjectures about the topology of the surfaces. Ioannis Roussos and I were able to verify the conjectures in our joint paper.

Dynamic Geometry of Polygons

Beginning with a cyclic polygon, Xin-Min Zhang and I (and later Jiu Ding) study the limiting behaviour of different sequences of related polygons constructed with iterative procedures. This has led to:

Sierpiński Pedal Triangles

This paper generalizes the classical construction of Sierpinski triangles to apply to pedal (orthic) triangles. Examples of the resulting 2-parameter family of fractals are given and various results about their fractal dimension are proven. The article just appeared in the physics journal Fractals.

A preliminary version of the paper (not the one to appear) is available here. An interactive java applet illustrating our construction of a sequence of pedal triangles is available for java-enabled browsers.

American Mathematical Monthly Problem Solution

When I was teaching a junior-level course in combinatorial enumeration in the spring 2003 semester, a problem appeared in the American Mathematical Monthly that related to the Stirling numbers we were studying. I prepared a long hand-out that lead up to the problem, and assigned it, along with several much easier, for homeowrk. Chad Versiga was able to solve the problem, and he and I submitted our combined solution to the Monthly. The solutions to the problem haven't appeared yet.

Future Projects