Claremont Topology Seminar: Jim Hoste (Pitzer College)

Speaker: Jim Hoste (Pitzer)

Title:Enumeration of Rational Knots, A New Proof of an Old Theorem

Abstract: I will begin with a quick review of rational knots, an interesting and well-studied class of knots. (Also known as 2-bridge knots.) In 1985, C. Ernst and D. W. Sumners gave an explicit formula for the number of rational knots with a given crossing number c. The formula is exponential in c, thus giving the first proof of the (widely believed) fact that the number of all prime knots grows exponentially with crossing number. In this talk I will give a new proof of their theorem which is diagrammatic in nature and which not only counts the number of rational knots with crossing number c, but also provides a fast and efficient way to tabulate these knots. These methods can also be used to calculate various invariants of rational knots and in some cases, the average value of a given invariant for a given crossing number.

This is joint work with Patrick D. Shanahan, Loyola Marymount University.

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We especially welcome all undergraduates and graduate students to attend topology seminar!

Room: Roberts South 103