A tribute to our former Associate Editor and practicioner of the experimental method.

We describe numerical experiments that suggest the existence of certain new compact surfaces of constant mean curvature. They come in three dihedrally symmetric families, with genus ranging from 3 to 5, 7 to 10, and 3 to 9, respectively; there are further surfaces with the symmetry of the Platonic polyhedra and genera 6, 12, and 30. We use the algorithm of Oberknapp and Polthier, which defines a discrete version of Lawson's conjugate surface method.

We use Brakke's Surface Evolver to deform a triply periodic minimal surface, the gyroid, into a continuous family of constant mean curvature surfaces with the same symmetry. We discuss stability and bifurcation problems for these surfaces.

There have been many attempts to settle the question whether there exist nontrivial 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 different HOMFLY, Jones and Alexander polynomials for knots up to 15 crossings.

We briefly discuss the relationship between several characterizations of the Hasse--Witt invariant of curves in characteristic $p$ with the goal of computing its value in concrete instances. We study its asymptotic behaviour when dealing with the geometric fibres of curves of genus $\geq 2$ defined over the rationals. Numerical evidence gathered for several modular curves supports certain conjectural distribution laws.

On the basis of experimental work involving matrix computations, we conjecture and prove that a criterion due to Bell for primeness of the universal enveloping algebra of a Lie superalgebra applies to the Cartan type Lie superalgebras $W(n)$ for $n=3$ but does not apply for odd $n\geq5$.

Motivated by problems from computer graphics and robotics---namely, ray tracing and assembly planning---we investigate the combinatorial structure of arrangements of segments on a line and of arcs on a circle. We show that there are, respectively, $1\times 3\times5\times\hbox{\mathversion{normal}$\cdots$}\times(2n{-}1)$ and $(2n) !$/$n!$ such arrangements; that the probability for the $i$-th endpoint of a random arrangement to be an initial endpoint is $(2n{-}i)$/$(2n{-}1)$ or $\half$, respectively; and that the average number of segments or arcs the $i$-th endpoint is contained in are $(i{-}1)(2n{-}i)$/$(2n{-}1)$ or $(n{-}1)$/$2$, respectively. The constructions used to prove these results provide sampling schemes for generating random inputs that can be used to test programs manipulating arrangements.

We also point out how arrangements are classically related to Catalan numbers and the ballot problem.