REU 1998: Polynomial Knots
The REU project for 1998 directed by Alan Durfee investigated polynomial
- Eva Curry (University of Maine)
- Eli Lebow (Harvard)
- Bryant Mathews (Simon's Rock/Harvard)
- Sang Pahk (Williams)
- Melanie Pivarski (Carnegie Mellon)
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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