We write $K_1 \geq K_2$ for two prime knots $K_1,K_2$ if there exists a surjective group homomorphism from $G(K_1)$ onto $G(K_2)$ where $G(K_1), G(K_2)$ are the knot groups of $K_1,K_2$, respectively. In this paper, we determine this partial order for the knots in Rolfsen's knot table.

We give bounds on the number of solutions to the Diophantine equation $(X+1/x)(Y+1/y) = n$ as $n$}tends to infinity. These bounds are related to the number of solutions to congruences of the form $ax+by \equiv 1$ modulo $xy$.

Let $M(P(z_1,\dots,z_n))$ denote Mahler's measure of the polynomial $P(z_1,\dots,z_n)$. Measures of polynomials in $n$ variables arise naturally as limiting values of measures of polynomials in fewer variables. We describe several methods for searching for polynomials in two variables with integer coefficients having small measure, demonstrate effective methods for computing these measures, and identify 48 polynomials $P(x,y)$ with integer coefficients, irreducible over $\Rats$, for which $1 < M(P(x,y)) < 1.37$.

We provide some statistics about an irreducibility/reducibility test for multivariate polynomials over finite fields based on counting points. The test works best for polynomials in a large number of variables and can also be applied to black-box polynomials.

We propose a cipher similar to the one-time pad and McEliece cipher based on a subband coding scheme. The encoding process is an approximation to the one-time pad encryption scheme. We present results of numerical experiments that suggest that a brute force attack on the proposed scheme does not result in all possible plaintexts, as the one-time pad does, but the brute force attack does not compromise the system. However, we demonstrate that the cipher is vulnerable to a chosen-plaintext attack.

If $K$ is a finite simplicial complex and $h$ is an injective map from the vertices of $K$ to $\R$, we show how to extend $h$ to a discrete Morse function in the sense of Forman in a reasonably efficient manner so that the resulting discrete Morse function mirrors the large-scale behavior of $h$. A concrete algorithm is given for the case where $K$ is a subcomplex of $\R^3$.

Holm studied modules of higher-order differential operators (generalizing derivations) on generic (central) hyperplane arrangements. We use his results to determine the Hilbert series of these modules. We also give a conjecture about the Poincaré-Betti series; these are known for the module of derivations through the work of Yuzvinsky and Rose and Terao.

This article presents an algorithm to compute Hilbert modular forms on the quadratic field $\Q(\sqrt{5})$. It also provides a list of all modular abelian varieties defined over $\Q(\sqrt{5})$ with prime level of norm less than 100 (up to $\Q$-isogeny).

We describe an experimental method for studying the combinatorics of Dirichlet domains in the complex hyperbolic plane based on numerical and graphical techniques. We apply our techniques to the complex reflection groups that appear in Mostow's seminal paper on the subject and list a number of corrections to the combinatorics of the Dirichlet domains.

We provide an algorithm for determining whether two vectors in the Leech lattice are equivalent under its isometry group, the Conway group $\co0$ of order $\sim8\times10^{18}$. Our algorithm reduces the test of equivalence to at most four tests under the subgroup $2^{12}{:}M_{24}$ and a test under this subgroup to at most 12 tests under $M_{24}$. We also give algorithms for testing equivalence under these two subgroups. We describe our intended applications to the symmetry groups of Lorentzian lattices and the enumeration of lattices of dimension ${}\sim24$ with good properties such as having small determinant. Our methods rely on and develop the work of R. T. Curtis.