In this paper, we present numerical computations of the winvariants and the Fintushel–Stern invariants for plumbed homology 3-spheres and use the results to test a conjecture of Witten suggesting that the invariants carry equivalent information. While the two invariants give nearly the same information for some homology 3-spheres, we present numerous examples in which the information carried by the two invariants is quite different.

We develop a method to find Rodrigues’s formulas for orthogonal matrix polynomials satisfying higher-order differential equations with coefficients independent of $n$.

In this paper we present two independent computational proofs that the monoid derived from 5 × 5 × 3 contingency tables is normal, completing the classification by Hibi and Ohsugi. We show that Vlach’s vector disproving normality for the monoid derived from 6 × 4 × 3 contingency tables is the unique minimal such vector up to symmetry. Finally, we compute the full Hilbert basis of the cone associated with the nonnormal monoid of the semigraphoid for |N| = 5. The computations are based on extensions of the packages LattE-4ti2 and Normaliz.

We give a method for verifying, by a symbolic calculation, the stability or semistability with respect to a linearization of fixed, possibly small, degree m, of the Hilbert point of a scheme $X ∈ \mathbb{P}(V)$ having a suitably large automorphism group.We also implement our method and apply it to analyze the stability of bicanonical models of certain curves. Our examples are very special, but they arise naturally in the log minimal model program for $\mathcal{M}_g$ . In some examples, this connection provides a check of our computations; in others, the computations confirm predictions about conjectural stages of the program.

We present new computations of approximately lengthminimizing polygons with fixed thickness. These curves model the centerlines of “tight” knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their knot types. While previous authors have minimized ropelength for polygons using simulated annealing, the new idea in our code is to minimize length over the set of polygons of thickness at least one using a version of constrained gradient descent.

We rewrite the problem in terms of minimizing the length of the polygon subject to an infinite family of differentiable constraint functions. We prove that the set of variations of a polygon of thickness one that does not decrease thickness to first order is a finitely generated polyhedral cone, and give an explicit set of generators. Using this cone, we give a first-order minimization procedure and a Karush–Kuhn–Tucker criterion for polygonalropelength criticality.

Our main numerical contribution is a set of 379 almost-critical knots and links, including all prime knots with ten and fewer crossings and all prime links with nine and fewer crossings. For links, these are the first published ropelength figures, and for knots they improve on existing figures. We give new maps of the self-contacts of these knots and links, and discover some highly symmetric tight knots with particularly simple-looking self-contact maps.

We point out that the elliptic genus of the K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group $M_{24}$. The reason remains a mystery.

We consider a substitution associated with the Arnoux–Yoccoz interval exchange transformation (IET) related to the tribonacci substitution.We construct the so-called stepped lines associated with the fixed points of the substitution in the abelianization (symbolic) space.We analyze various projections of the stepped line, recovering the Rauzy fractal, a Peano curve related to work in Arnoux, another Peano curve related to the work of Mc- Mullen and Lowenstein et al., and also the interval exchange transformation itself.