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



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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