To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The Volume Conjecture for small angles states that the value of the n-th colored Jones polynomial at eα/n is a sequence of complex numbers that grows subexponentially, for a fixed small complex angle α. In an earlier publication, the authors proved the Volume Conjecture...

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

Let K be a knot in the three-sphere and JN (K; t) the colored Jones polynomial corresponding to the N -dimensional representation of sl2(C) normalized so that JN (unknot; t) = 1 [8, 12]. R. Kashaev found a series of link invariants parameterized by positive integers [9] and proposed a conjecture that the asymptotic behavior of his invariants would determine the hyperbolic volume of the knot com...