Susan G. Williams

Toeplitz minimal flows which are not uniquely ergodic

SummaryWe study minimal symbolic dynamical systems which are orbit closures of Toeplitz sequences. We construct 0–1 subshifts of this type for which the set of ergodic invariant measures has any given finite cardinality, is countably infinite or has cardinality of the continuum.

Mahler measure, links and homology growth

Abstract Let l be an oriented link of d components with nonzero Alexander polynomial Δ(u1,…,ud). Let Λ be a finite-index subgroup of H 1 (S 3 −l)≅ Z d , and let MΛ be the corresponding abelian cover of S3 branched along l. The growth rate of the order of the torsion subgroup of H1(MΛ), as a suitable measure of Λ approaches infinity, is equal to the Mahler measure of Δ.

ALEXANDER GROUPS AND VIRTUAL LINKS

The extended Alexander group of an oriented virtual link l of d components is defined. From its abelianization a sequence of polynomial invariants Δi (u1,…,ud, v), i=0, 1,…, is obtained. When l is a classical link, Δi reduces to the well-known ith Alexander polynomial of the link in the d variables u1v,…,udv; in particular, Δ0 vanishes.

Determinants of Commuting-Block Matrices

Let R be a commutative ring, and let Matn(3W) denote the ring of n x n matrices over S. We can regard a k x k matrix M= (A(D) over Matn(R) as a block matrix, a matrix that has been partitioned into k2 submatrices (blocks) over M, each of size n x n. When M is regarded in this way, we denote its determinant in R by IMI. We use the symbol D(M) for the determinant of M viewed as a k x k matrix over Matn(W). It is important to realize that D(M) is an n x n matrix.

Mahler Measure of Alexander Polynomials

Let l be an oriented link of d components in a homology 3-sphere. For any nonnegative integer q, let l(q) be the link of d-1 components obtained from l by performing 1/q surgery on the dth component. Then the Mahler measure of the Alexander polynomial of l(q) converges to the Mahler measure of the Alexander polynomial of l as q goes to infinity, provided that some other component of l has nonzero linking number with the dth. Otherwise, the Mahler measure of the Alexander polynomial of l(q) has a well-defined bu different limiting behavior. Examples are given of links for which the Mahler measure of the Alexander polynomial is small. Possible connection with hyperbolic volume are discussed.

Knot invariants from symbolic dynamical systems

If G is the group of an oriented knot k, then the set Hom(K, J) of representations of the commutator subgroup K = [G, G] into any finite group E has the structure of a shift of finite type (D, a special type of dynamical system completely described by a finite directed graph. Invariants of (D, such as its topological entropy or the number of its periodic points of a given period, determine invariants of the knot. When S is abelian, (r gives information about the infinite cyclic cover and the various branched cyclic covers of k. Similar techniques are applied to oriented links.

POLYNOMIAL INVARIANTS OF VIRTUAL LINKS

Properties of polynomial invariants Δi for oriented virtual links are established. The effects of taking mirror images and reversing orientation of the link diagram are described. The relationship between Δ0(u,v) and an invariant of F. Jaeger, L. Kauffman, H. Saleur and J. Sawollek is discussed.

On reciprocality of twisted Alexander invariants

Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for which the twisted Reidemeister torsion is not reciprocal.

Euclidean Mahler measure and twisted links

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2-variable Homflypt polynomials of the corresponding links have bounded Mahler measure.

VIRTUAL KNOT GROUPS

Virtual knot groups are characterized, and their properties are compared with those of classical knot groups. A coloring theory, generalizing the usual notion of Fox n-coloring, is introduced for virtual oriented links.