The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none of which is the automorphic image of another, such that each normally generates the group.

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n, 1), n > 0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under free group endomorphisms.

We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We also give conditions for expansivity of the quotient, and applications to combinatorial group theory, knot theory and topological quantum field theory.

Given a compact, connected, oriented 3-manifold $M$ with boundary, and epimorphism $\chi$ from $H_1M$ to a free abelian group $\Pi$, two invariants $\beta$, $\tau \in \bb {Z}\Pi$ are defined. If $M$ embeds in another such 3-manifold $N$ such that $\chi_N$ factors through $\chi$, then the product $\beta\tau$ divides $\Delta_0(H_1\tilde {N})$.

A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein theoretic generalizations for $2n$-tangles provide invariants that persist in the corresponding invariants of links in which they embed. An example is given of a virtual 4-tangle for which Krebes's theorem does not hold.

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 its $d$th component $l_d$. The Mahler measure of the multivariable Alexander polynomial $\Delta_{l(q)}$ converges to the Mahler measure of $\Delta_l$ as $q$ goes to infinity, provided that $l_d$ has nonzero linking number with some other component. If $l_d$ has zero linking number with each of the other components, then the Mahler measure of $\Delta_{l(q)}$ has a well defined but different limiting behavior. Examples are given of links $l$ such that the Mahler measure of $\Delta_l$ is small. Possible connections with hyperbolic volume are discussed.

Given a finitely presented group G and an epimorphism χ:G→Z, constraints on the orders of automorphisms F:G→G such that χ∘F = χ are obtained via symbolic dynamics. The techniques provide new obstructions to periodicity for knots and links.

For n ≥ 3, an n-knot K has a minimal Seifert manifold if and only if its group is isomorphic to an HNN-extension with finitely presented base. In this case, any Seifert manifold for K can be converted to a minimal Seifert manifold for K by some finite sequence of ambient 0- and 1-surgeries.

We work throughout in the smooth category. Homomorphisms of fundamental and homology groups are induced by inclusion. An n-knot, form n ≥ 1, is an embedded n-sphere K ⊂ Sn+2. A Seifert manifold for K is a compact, connected, orientable (n + 1)-manifold V ⊂ Sn+2 with boundary ∂V = K. By [9] Seifert manifolds always exist. As in [9] let Y denote Sn+2 split along V; Y is a compact manifold with ∂Y = V0 ∪ V1, where Vt ≈ V. We say that V is a minimal Seifert manifold for K if π1Vt → π1Y is a monomorphism for t = 0, 1. (Here and throughout basepoint considerations are suppressed.)

We determine the effect on the Jones polynomial evaluated at t = i and t = eπi/3 of an oriented link whenever certain twists are performed.

We investigate the effect on the Jones polynomial of a ribbon knot when two of its bands are twisted together. We use our results to prove that each of the three S-equivalence classes of genus 2 fibered doubly slice knots in S3 can be represented by infinitely many distinct prime fibered doubly slice ribbon knots.