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Trip graphs of classical knots
by
Lorenzo Traldi
Lafayette College
Coauthors: Louis Zulli (Lafayette College)
A plane diagram of a classical knot yields a chord diagram, which in turn yields an intersection graph. If we modify this intersection graph by adjoining loops at the vertices that correspond to negative crossings, we obtain the "trip graph". Its adjacency matrix is the trip matrix, which yields the Jones polynomial [Zulli, Topology 34 (1995)]. This derivation of the Jones polynomial is quite different from the better-known derivation from the checkerboard graph, related to the Tutte polynomial [Thistlethwaite, Topology 26 (1987)]. Crossings give rise to edges in the checkerboard graph and vertices in the trip graph, for one thing. Also, the trip graph construction does not extend to links of more than one component, so it does not give rise to the usual recursive description of the Jones polynomial.
We discuss the invariants of arbitrary graphs that correspond to the Kauffman bracket and the Jones polynomial. They have several interesting properties. For instance: the graph bracket distinguishes nonisomorphic looped graphs with no more than 6 vertices, and also simple graphs with 7 vertices; the graph bracket may be calculated through an unfamiliar recursion that uses the pivot and local complementation operations associated with the interlace polynomial [Arratia, Bollobás and Sorkin, J. Combin. Theory Ser. B 92 (2004)]; and the graph Jones polynomial is invariant under graph-theoretic versions of the Reidemeister moves.
Date received: February 21, 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 # cawt-04.