and animated man with a dazzling set of credentials, including over 18 500 citations of his research output. What I immediately liked about him was his pragmatic determination to improve the quality of clinical performance by whatever means comes to hand. Ian Jones does not see himself as a purist researcher: 'The ultimate goal of research is making things better for people with these condition...

Circulants over Finite Fields The most basic question we can ask in the study of circulants over a finite fields is the structure of the units in the circulant ring. With one exception, we shall provide a complete description of the isomorphism class of the unit group. But, we shall only provide a method of constructing these unit groups assuming generators to the units of various fields are gi...

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 ...