# REU 2004: 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

- Elizabeth Bellenot, Wellesley College
- Ashley Brown, University of California, Berkeley
- William Espenschied, University of Central Florida
- Wing Leung Mui, Amherst College
- Colleen Sweeney, Fairfield University

## 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:

- Ashley Brown,
*Examples of polynomial knots*(pdf)

** Abstract: ** In this paper, we define and give
examples of polynomial knots. In particular, we write down specific
polynomial equations with rational coefficients for seven different
knots, ranging from the figure eight knot to a knot with ten
crossings. A nice property that all of these knots share is that they
have planar projections that are symmetric about the y-axis.

- Wing Mui,
*Completing a polynomial knot from a projection*(pdf)

** Abstract: ** Given a knot type and a
projection of it parameterized by two polynomials, we can find a
third polynomial that will form a representation of the knot in three
space along with the two given polynomials. This paper discusses
methods with which to find this third polynomial and some related
results.

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