Braids: A Survey

This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labelled, as we encounter them.Comment: Final version, revised to take account of the comments of readers. A review article, to appear in the Handbook of Knot Theory, edited by W. Menasco and M. Thistlethwaite. 91 pages, 24 figure

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

Colloquium : disclination loops, point defects, and all that in nematic liquid crystals

The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress was made in controlling and measuring colloidal inclusions in liquid crystalline phases. The topological structure of these systems is quite rich but, at the same time, subtle. Motivated by experiment and the power of topological reasoning, the classification of defects in uniaxial nematic liquid crystals was reviewed and expounded upon. Particular attention was paid to the ambiguities that arise in these systems, which have no counterpart in the much-storied XY model or the Heisenberg ferromagnet

Knots, Trees, and Fields: Common Ground Between Physics and Mathematics

One main theme of this thesis is a connection between mathematical physics (in particular, the three-dimensional topological quantum field theory known as Chern-Simons theory) and three-dimensional topology. This connection arises because the partition function of Chern-Simons theory provides an invariant of three-manifolds, and the Wilson-loop observables in the theory define invariants of knots. In the first chapter, we review this connection, as well as more recent work that studies the classical limit of quantum Chern-Simons theory, leading to relations to another knot invariant known as the A-polynomial. (Roughly speaking, this invariant can be thought of as the moduli space of flat SL(2,C) connections on the knot complement.) In fact, the connection can be deepened: through an embedding into string theory, categorifications of polynomial knot invariants can be understood as spaces of BPS states. We go on to study these homological knot invariants, and interpret spectral sequences that relate them to one another in terms of perturbations of supersymmetric theories. Our point is more general than the application to knots; in general, when one perturbs any modulus of a supersymmetric theory and breaks a symmetry, one should expect a spectral sequence to relate the BPS states of the unperturbed and perturbed theories. We consider several diverse instances of this general lesson. In another chapter, we consider connections between supersymmetric quantum mechanics and the de Rham version of homotopy theory developed by Sullivan; this leads to a new interpretation of Sullivan's minimal models, and of Massey products as vacuum states which are entangled between different degrees of freedom in these models. We then turn to consider a discrete model of holography: a Gaussian lattice model defined on an infinite tree of uniform valence. Despite being discrete, the matching of bulk isometries and boundary conformal symmetries takes place as usual; the relevant group is PGL(2,Qp), and all of the formulas developed for holography in the context of scalar fields on fixed backgrounds have natural analogues in this setting. The key observation underlying this generalization is that the geometry underlying AdS3/CFT2 can be understood algebraically, and the base field can therefore be changed while maintaining much of the structure. Finally, we give some analysis of A-polynomials under change of base (to finite fields), bringing things full circle.</p