REU 2006: Polynomial Knots

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

Student participants

• Susan Durst
• Emily List
• Charles Siegel
• Katherine Toman
• Matthew Wright

Brief Description

A polynomial knot is a polynomial map R -> 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.

Reports

The group produced the following papers:

Susan Durst, The parameter space of polynomial knots of degree three (pdf)

Abstract: We prove that the space of polynomial knots of degree 3 is star-like, and hence contractible. (NB: This paper later developed into a senior thesis.)

Emily List, Polynomial knots with the same knot diagram

Abstract: We find polynomial equations for the knots with crossing number eight with the same knot projection.

Charles Siegel, Counting singularities of parametric curves (pdf)

Abstract: In this paper, we consider polynomial parametrized curves in the affine plane $k^2$ over an algebraically closed field $k$. Such curves are given by $\kappa:k\to k^2:t\mapsto(x(t),y(t))$, and may or may not contain singular points. The problem of how many singular points there are is of specific importance to the theory of polynomial knots, as it gives a bound on the degrees necessary to achieve a parametrization of a knot with a specified number of crossings. We will state a more general conjecture and supply a proof of it in a special case.

Kathrine Toman, Polynomial knots of six crossings or less (pdf)

Abstract: The knot types of some equations from Brown (REU 04) are identified.

Matthew Wright, Constructing polynomial knots (pdf)

Abstract: We present an algorithm for converting a piecewise linear parameterization of a knot into a polynomial parameterization of the same knot type, and prove that this is so. We also show how this algorithm can be modified to give compact knots, and (in some cases) trigonometric knots.