Does the Jones Polynomial Detect the Unknot?

Does the Jones Polynomial Detect the Unknot ? Stephen

A cabling formula for the colored Jones polynomial

We prove an explicit cabling formula for the colored Jones polynomial. As an application we prove the volume conjecture for all zero volume knots and links, i.e. all knots and links that are obtained from the unknot by repeated cabling and connected sum.

2 00 3 Categorifications of the colored Jones polynomial

The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n + 1], the quantum dimension of the (n + 1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.

Asymptotic Behaviors of the Colored Jones Polynomials of a Torus Knot

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...

Any Tangle Extends to Non-mutant Knots with the Same Jones Polynomial

The following question is still unanswered: Is there a nontrivial knot with Jones polynomial 1? This question has motivated a range of tools for generating pairs of distinct knots sharing a common Jones polynomial [1, 7, 14, 15]. The prototype for these is the construction of mutation due to Conway [4], and it is well known that this technique preserves the HOMFLY polynomial [7]. However, it ca...