# 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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