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)|
|Time:||9 am (Session I)|