REU 2002: Polynomial and Rational Knots

This REU project was directed by Alan Durfee and Donal O'Shea of Mount Holyoke College. The topic of investigation was polynomial and rational parameterizations of knots.

Student participants

(The picture above is of both REU groups.)

David Clark, Michigan Technical University '04
Donovan McFeron, University of Notre Dame '03
Virginia Peterson, University of Massachusetts, Amherst '03
Craig Phillips, Rutgers University '03
Alexandra Zuser, Marlboro College '03

Brief Description

A polynomial knot is a polynomial map from R to R^3 which is a smooth embedding. These were first looked at by Shastri (Tohoku Math. J. 44 (1992) 11-17) and then by Vassiliev. Polynomial knots do not have compact image. If the parameterization is rational (a rational knot), then the image can be compact. This group investigated both polynomial and rational knots.

Reports

The group produced the following papers:

David Clark, Transforming trigonometric knot parameterizations into rational knot parameterizations (pdf): Abstract: This paper develops a method for constructing rational parameterizations of knots, based on a trigonometric parameterization, with explicit results for torus knots.
Donovan McFeron Algorithm for construction compact rational knots (pdf): Abstract: This paper defines an algorithm for constructing compact rational knots for a given knot type. A trefoil folowing this algorithm is constructed.
Donovan McFeron The minimal degree sequence of the polynomial figure-eight knot (pdf): Abstract: This paper defines the minimal degree sequence of a polynomial knot. It is determined for the figure-eight knot and equations for a parameterization of degree 6 are given.
Donovan McFeron and Alexandra Zuser On the degrees of rational knots (pdf): Abstract: Minimal degrees for parameterizations of rational knots are investigated.
Virginia Peterson Mirror images of knots: Abstract: A polynomial parameterization of some knots is converted to a compact rational parameterization of the mirror image.
Craig Phillips Three methods of constructing rational representations of knots (pdf): Abstract: This paper will discuss different ways of finding rational parameterizations of knots in three-space. We will first show a way of constructing torus knots, and then two ways of converting polynomial representatings of knots into rational representations.
Alexandra Zuser Distinguishing knots with the Alexander Polynomial (postscript): Abstract: This paper contains an algorithm for determining purely algebraically the knot type of a polynomial or rational knot of low crossing number.

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