Anyons in Geometric Models of Matter

We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.Comment: 22 pages LaTe

Knots on a positive template have a bounded number of prime factors

Templates are branched 2-manifolds with semi-flows used to model `chaotic' hyperbolic invariant sets of flows on 3-manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be bounded. We prove a special case when the template is positive; the general case is now known to be false.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-24.abs.htm

Dynamical ideals of non-commutative rings

An analog of the prime ideals for simple non-commutative rings is introduced. We prove the fundamental theorem of arithmetic for such rings. The result is used to classify the surface knots and links in the smooth 4-dimensional manifolds.Comment: 11 page

Simple Smale flows and their templates on S3S^3

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of $3$-dimensional flows. We prove that for an embedded template in $S^3$ with fixed homeomorphism type, its boundary as a trivalent spatial graph is a complete isotopic invariant. Moreover, we construct an invariant of embedded templates by Kauffman's invariant of spatial graphs, which is a set of knots and links. As application, the isotopic classification of simple Smale flows on $S^3$ is discussed.Comment: 14 pages, 3 figure

Tangles, Generalized Reidemeister Moves, and Three-Dimensional Mirror Symmetry

Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch lines collide. Massive deformations of the field theory correspond to resolutions of singularities resulting in distinct smooth manifolds connected by geometric transitions. A generalization of Reidemeister moves for singular tangles captures mirror symmetries of the underlying theory yielding a geometric framework where dualities are manifest.Comment: 80 pages, 48 figure

Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Global analysis of dynamical systems on low-dimensional manifolds.

The interaction of topology and dynamics has attracted a great deal of attention from numerous mathematicians. This thesis is devoted to the study of dynamical systems on low-dimensional manifolds. In the order of dimensions, we first look at the case of two-manifolds (surfaces) and derive explicit differential equations for dynamical systems defined on generic surfaces by applying elliptic and automorphic function theory to uniformise the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series, we will determine general meromorphic systems on a fundamental domain in the upper half plane, the solution trajectories of which 'roll up' onto an appropriate surface of any given genus. Meanwhile, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by analyzing the topology of inversely unstable solutions contained within each invariant set. Then the thesis concerns a study of three-dimensional systems. We give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tubular neighbourhood of the braid. The second one, generating chaotic systems, is realized by modelling the Smale horseshoe. Moreover, we shall consider the analytical and topological structure of systems on 2- and 3- manifolds. By considering surgery operations, such as Dehn surgery, Heegaard splittings and connected sums, we shall show that it is possible to obtain systems with 'arbitrarily strange' behaviour, Le., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We will also consider diffeomorphisms which are defined on closed 3-manifolds and contain generalized Smale solenoids as the non-wandering sets. Motivated by the result in [Jiang, Ni and Wang, 2004], we will investigate the possibility of generating dynamical systems containing an arbitrary number of solenoids on any closed, orientable 3-manifold. This shall also include the study of branched coverings and Reeb foliations. Based on the intense development from four-manifold theory recently, we shall consider four-dimensional dynamical systems at the end. However, this part of the thesis will be mainly speculative