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Riley Polynomials of 2-Bridge Knots
by
Susan Williams
University of South Alabama
Coauthors: Daniel Silver

Two-bridge knots are parameterized by pairs of relatively prime integers (α, β), 0 < β < α, with at most two pairs determining the same knot type k(α, β). R. Riley described a procedure for associating to each such pair a polynomial Φ_{α, β} with roots corresponding to the nonabelian parabolic SL₂C representations of the knot group. (The representation is said to be parabolic if the image of any meridian has trace 2.)

We survey known results about this Riley polynomial, and present some new ones. In particular, if Φ_{α, β} is composite then at least one irreducible factor is not the Riley polynomial of a 2-bridge knot. If exactly one factor φ of Φ_{α, β} is not the Riley polynomial of a 2-bridge knot, and k(α, β) is not a torus knot, then the roots of φ correspond to faithful representations.

Date received: December 2, 2008

Copyright © 2008 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 # caxq-53.