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Polynomial and Rational knots
by
Alan Durfee
Mount Holyoke College
Coauthors: Don O'Shea, Mount Holyoke College
Abstract: A polynomial knot is defined to be a smooth embedding of R1 in R3 whose components are real polynomials. For example, the map (t5 - 10t, t4 - 4t2, t3 - 3t) represents a trefoil knot (Shastri, 1992). These knots have ends going out to infinity. Mishra, Mount Holyoke REU groups and others have found many examples of polynomial knots. Similarly a rational knot in real projective three-space is defined by rational equations. A polynomial knot completes to a rational knot, and there are many other examples.
One can show that a topological knot can be approximated by a polynomial knot of the same knot type. Given a topological knot, there are no good lower bounds for the degree of a polynomial approximation, nor are there methods for constructing examples of low degree, though past work has clarified these questions to some extent. Also the topological structure of the space of polynomial knots of a given degree is not known, though Vassiliev found this for degrees at most 4, and in degree 5 this question was partially solved by the 2002 MHC REU group.
Current research will be described and problems for future work will be indicated.
Date received: January 19, 2005
Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capo-06.