A trace formula for the forcing relation of braids

The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.Comment: 24 pages, 9 figure

The index of coincidence Nielsen classes of maps between surfaces

AbstractFor a given pair of closed orientable surfaces Sh, Sg and given integers d1, d2, one would like to find bounds for the index of the Nielsen coincidence classes among all possible pairs of maps (f1,f2):Sh→Sg where |deg(f1)|=d1 and |deg(f2)|=d2. We show that these bounds are infinite when h>g=1, or when h⩾g>1 and both di<(h−1)/(g−1). We calculate these bounds when h=g and d2=1. We also consider the similar question for the root case, which is simpler, and we solve it completely. Few results are given when di=(h−1)/(g−1) for either i=1 or i=2

3-manifolds that admit knotted solenoids as attractors

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S^3 has Property IE if the infinite cyclic cover of the knot exterior embeds into S^3. Clearly all fibred knots have Property IE. There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δk(t) must be either 1 or 2t^2−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having _(Δk)(t)=1 and 2t^2−5t+2 respectively. Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings. We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold