Using the principle of symmetric criticality [Palais 1979], we construct torus knots and links that extremize the Möbius-invariant energy introduced by O'Hara [1991] and Freedman, He and Wang [1993]. The critical energies are explicitly computable using the calculus of residues, a result obtained in collaboration with Gil Stengle.

Experiments with a discretized version of the Möbius energy--applicable to the study of arbitrary knots and links--are also described, and confirm the results of the analytic calculations.

We give an explicit construction for the two faithful, irreducible, 16-dimensional representations of $4\udot M_{22}$ over the field $\GF(49)$. Then we extend them to the 32-dimensional representation of $4\udot M_{22}{:}2$ over $\GF(7)$. Explicit matrices are given on page 14.

We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in $\R^3$, $\Sph ^3$ and $\H^3$. The algorithm makes no restriction on the genus and can handle singular triangulations.

Additionally, we present an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map. This can be applied to the identity map on a minimal surface to produce its conjugate minimal surface, a procedure that often yields unstable solutions to a free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.

This article is an expository description of quadratic rational maps from the Riemann sphere to itself.