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Scholars Day: April 15, 2009

A Computational Investigation: Does the Jones Polynomial Detect Unknotedness?

A famous conjecture in knot theory states that there are no nontrivial knots with unit Jones polynomial. In this work, knot diagrams are enumerated using a method similar to that of Yamada: decomposition into Conway polyhedra (simple 4-valent planar graphs) on which each vertex contains an algebraic tangle, and computation, for each knot, of a quantity proportional to the Jones polynomial. There is a combinatorial explosion in computational effort with number of crossings. However, 8% of Conway polyhedra lead to “connected sum” knots and can thereby be eliminated from consideration; significant numbers of computing functions can be eliminated by a signed permutation representation scheme; and many algebraic tangles can be eliminated by considering wave moves. Furthermore, almost every polynomial computation can be replaced by a single floating point computation, resulting in order of magnitude decreases in memory requirements and computation time. As of this writing, no nontrivial knots with unit Jones polynomials have been found.

Presenter: Robert Tuzun (Faculty)
Topic: Computational Science
Location: 125 Hartwell
Time: 9 am (Session I)