Non-geometric veering triangulations

Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic metric with all tetrahedra positively oriented. This paper describes a computer program Veering, building on the program Trains by Toby Hall, for generating these triangulations starting from a description of the homeomorphism as a product of Dehn twists. Using this we obtain the first examples of non-geometric veering triangulations; the smallest example we have found is a triangulation with 13 tetrahedra

Roots, symmetries and conjugacy of pseudo-Anosov mapping classes

An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: decide if the element commutes with a finite order element and in this case compute this element. The structure theorem on which this algorithm is based provides also a new solution to the conjugacy problem

Nielsen theory, braids and fixed points of surface homeomorphisms

AbstractWe study two problems in Nielsen fixed point theory using Artin's braid groups and the Nielsen–Thurston classification of surface homeomorphismsup to isotopy. The first is that of distinguishing Reidemeister classes of free group automorphisms realized by a braid (and thus induced by homeomorphismsof the 2-disc relative to a finite invariant set), for which we give a necessary and sufficient condition in terms of a conjugacy problem in the braid group. Consequently, one may use any braid conjugacy invariant (those of Garside's algorithm, linking numbers, topological entropy, etc.) and any link invariant (Alexander polynomial, splittability, etc.) to distinguish Reidemeisterclasses, giving much stronger criteria than those already known.The second problem is that of deciding when two fixed points of a surface homeomorphismbelong to the same Nielsen fixed point class. We give two criteria, the first in terms of certain reducing curves which can be checked using the Bestvina–Handel algorithm, the second using the multi-variable Alexander polynomial of a link associated with the suspension of the homeomorphism.Finally we consider generalizations of Sharkovskii's theorem on the coexistence of periodic orbits of interval maps to homeomorphismsof the 2-disc. We show that for each n⩾5 there exists a pseudo-Anosovorientation-preservinghomeomorphismof the 2-disc relative to a periodic orbit of period n that does not have periodic orbits of all periods, with an analogous result for the 2-sphere

Topological fluid mechanics of point vortex motions

Topological techniques are used to study the motions of systems of point vortices in the infinite plane, in singly-periodic arrays, and in doubly-periodic lattices. The reduction of each system using its symmetries is described in detail. Restricting to three vortices with zero net circulation, each reduced system is described by a one degree of freedom Hamiltonian. The phase portrait of this reduced system is subdivided into regimes using the separatrix motions, and a braid representing the topology of all vortex motions in each regime is computed. This braid also describes the isotopy class of the advection homeomorphism induced by the vortex motion. The Thurston-Nielsen theory is then used to analyse these isotopy classes, and in certain cases strong conclusions about the dynamics of the advection can be made

Topological Chaos in Spatially Periodic Mixers

Topologically chaotic fluid advection is examined in two-dimensional flows with either or both directions spatially periodic. Topological chaos is created by driving flow with moving stirrers whose trajectories are chosen to form various braids. For spatially periodic flows, in addition to the usual stirrer-exchange braiding motions, there are additional topologically-nontrivial motions corresponding to stirrers traversing the periodic directions. This leads to a study of the braid group on the cylinder and the torus. Methods for finding topological entropy lower bounds for such flows are examined. These bounds are then compared to numerical stirring simulations of Stokes flow to evaluate their sharpness. The sine flow is also examined from a topological perspective.Comment: 18 pages, 14 figures. RevTeX4 style with psfrag macros. Final versio

Algorithmic detectability of iwip automorphisms

We produce an algorithm that, given $\phi\in Out(F_N)$, where $N\ge 2$, decides wether or not $\phi$ is an iwip ("fully irreducible") automorphism.Comment: final version, to appear in the Bulletin of London Math. So