This result has an application to the hyperbolic subset of the p-adic Mandelbrot set, whose complex analogue has been much studied [2, 4, 5, e.g.]. Much of the proof of (2) is an analysis of the Galois tower formed by the splitting fields of iterates of y2+x ∈ Fp(x)[y]. Similar towers have been studied recently by Morton [7], Odoni [8], and Aitken, Hajir, and Maire [1]. I introduce a stochastic...

From the braid-valued Burau module over the braid group we construct the Yang-Baxter matrices yielding the Alexanderand the Jones knot invariants. This generalises an observation of V. F. R. Jones.

There were many attempts to settle the question whether there exist non-trivial knots with trivial Jones polynomial. In this paper we show that such a knot must have crossing number at least 18. Furthermore we give the number of prime alternating knots and an upper bound for the number of prime knots up to 17 crossings. We also compute the number of diierent Hommy, Jones and Alexander polynomia...

A link is a finite family of disjoint, smooth, oriented or unoriented, closed curves in R or equivalently S. A knot is a link with one component. The Jones polynomial VL(t) is a Laurent polynomial in the variable √ t which is defined for every oriented link L but depends on that link only up to orientation preserving diffeomorphism, or equivalently isotopy, of R. Links can be represented by dia...

From the braid-valued Burau module over the braid group we construct the Yang-Baxter matrices yielding the Alexander-and the Jones knot invariants. This generalises an observation of V. F. R. Jones.

Abstract. R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots 63, 89 and 820 and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern–Simons ...

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...