REU 1998: Polynomial Knots

The REU project for 1998 directed by Alan Durfee investigated polynomial 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. Before this summer only a few examples of polynomial knots were known. The group computed many more examples, including an algorithm for the connected sum of two knots and an infinite family of knots in minimal degree. They also found a lower bound for the degree of the polynomial in terms of Kuiper's superbridge number, thus extending work of Rudolph who found a lower bound in terms of the crossing and bridge numbers.

Durfee presented these results at the 1998 ICM in Berlin; his lecture transparencies (in Tex) provide a general report on the summer's activities.


The group produced the following papers:

Bryant Mathews, Determining the knot type of a polynomial knot (in Tex).

Abstract: An algorithm is presented for calculating the crossing data for a polynomial knot

And more to come!

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