Topological Entropy of Braids on the Torus

A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir's technique for braids on the sphere. Previous methods for computing topological entropies include the Bestvina--Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina--Handel algorithm quickly becomes computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and appears to give exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow. The efficiency of the method allows us to explore how much extra information about flow entropy is encoded in the braid as the number of trajectories becomes large.Comment: 19 pages, 44 figures. SIAM journal styl

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

Braids, Conformal Module and Entropy

The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conformal module is inverse proportional to a popular dynamical braid invariant, the entropy. The entropy appeared in connection with Thurston's theory of surface homeomorphisms. An application of the concept of conformal module to algebraic geometry is given.Comment: Research announcement, 4 pages Minor revision, misprints correcte

Estimating topological entropy from the motion of stirring rods

Stirring a two-dimensional viscous fluid with rods is often an effective way to mix. The topological features of periodic rod motions give a lower bound on the topological entropy of the induced flow map, since material lines must `catch' on the rods. But how good is this lower bound? We present examples from numerical simulations and speculate on what affects the 'gap' between the lower bound and the measured topological entropy. The key is the sign of the rod motion's action on first homology of the orientation double cover of the punctured disk.Comment: 10 pages, 20 figures. IUTAM Procedia style (included). Submitted to volume "Topological Fluid Dynamics II.

On the stability of topological phases on a lattice

We study the stability of anyonic models on lattices to perturbations. We establish a cluster expansion for the energy of the perturbed models and use it to study the stability of the models to local perturbations. We show that the spectral gap is stable when the model is defined on a sphere, so that there is no ground state degeneracy. We then consider the toric code Hamiltonian on a torus with a class of abelian perturbations and show that it is stable when the torus directions are taken to infinity simultaneously, and is unstable when the thin torus limit is taken

Fluid Stirring on a Sphere -- a Topological Approach

The stirring and mixing of a fluid with moving rods is vital in many physical applications in order to achieve homogeneity within the mixture. These rods act as an obstacle that stretches and folds together fluid elements. Over time, the permutation of these rods comprise a mathematical braid whose properties dictate a minimum topological entropy, a number to describe the total disorder or chaos of a system. A braid whose topological entropy is greater than one exhibits chaotic behavior which guarantees an optimal mixing pattern. These rod stirring braids have been previously studied on both the disk as well as the two dimensional torus. The trajectory of fluid mixing on a sphere poses an intriguing starting inquiry to overall mixing on spherical surfaces like the ocean, stars, etc. We use a recipe established by Yvon Verberne to create pseudo-Anosov maps on a punctured sphere using Dehn twist in order to construct similar maps on a 4-times punctured sphere. Since a quotient of the torus under a hyperelliptic involution of torus is the 2-sphere with 4 marked points, we are able to use various methods in order to estimate the topological entropy of a stirring protocol on a 4-times punctured sphere