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 with as many full twists as strands realize the braid index

We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function-valued concordance homomorphism defined by Ozsv\'ath, Stipsicz, and Szab\'o. We use this characterization to prove that $n$-braids with fractional Dehn twist coefficient larger than $n-1$ realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that $n$-times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.Comment: 26 pages, 5 figures, comments welcome! V2: Implementation of referee suggestions. Accepted for publication by the Journal of Topolog

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

Integration and conjugacy in knot theory

This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum. There is a folklore result in physics saying that the Feynman integration of an exponential is itself an exponential. In chapter 2 we state and prove an exact formulation of this statement in the language which is used in the theory of finite type invariants. The final chapter is concerned with properties of link groups. In particular we study the relationship between known solutions from small cancellation theory and normal surface theory for the word and conjugacy problems of the groups of (prime) alternating links. We show that two of the algorithms in the literature for solving the word problem, each using one of the two approaches, are the same. Then, by considering small cancellation methods, we give a normal surface solution to the conjugacy problem of these link groups and characterize the conjugacy classes. Finally as an application of the small cancellation properties of link groups we give a new proof that alternating links are non-trivial.Comment: University of Warwick Ph.D. thesi

Braiding transitions and plectonemic structures in multiple-stranded chains manipulated by magnetic tweezers

openThanks to the recent advance in micromanipulation techniques based for instance on optical and magnetic tweezers, it is nowadays possible to probe the mechanical response and the configurational transitions of soft structures made by multiple linear polymers, such as ds-DNA filaments, that wrap one to another in a braided fashion. In particular, by using magnetic tweezers one can look at the braided/plectonemic (or buckling) transition of these structures as a function of the extensional force and torsion injected on the system. Recent theoretical and experimental studies have focused on structures made by only two filaments. The aim of this thesis is to extend these investigations to the case of multiple (i.e. more then two) strands where the reciprocal position of the rooted monomers at the tweezeers' plates and the detection of plectonemic structures are interesting novel issues to be explored. Geometric quenches between three-stranded and two-stranded configurations are also explored by introducing a cut along the additional third strand and simulating the system relaxing to equilibrium. The analytical approach is based on the elastic rod model of a chain with bend and twist rigidities, while numerical simulation are performed on a coarse-grained model of three stranded chains whose stochastic dynamics is integrated using LAMMPS code. The study of such new configurations highlights the presence of a buckling transition similar to the one found with two strands, in which the coexistence of plectonemic and non formations is more pronounced than what previously observed. For such phase transition the geometric properties of the system influence directly the critical points positioning.Thanks to the recent advance in micromanipulation techniques based for instance on optical and magnetic tweezers, it is nowadays possible to probe the mechanical response and the configurational transitions of soft structures made by multiple linear polymers, such as ds-DNA filaments, that wrap one to another in a braided fashion. In particular, by using magnetic tweezers one can look at the braided/plectonemic (or buckling) transition of these structures as a function of the extensional force and torsion injected on the system. Recent theoretical and experimental studies have focused on structures made by only two filaments. The aim of this thesis is to extend these investigations to the case of multiple (i.e. more then two) strands where the reciprocal position of the rooted monomers at the tweezeers' plates and the detection of plectonemic structures are interesting novel issues to be explored. Geometric quenches between three-stranded and two-stranded configurations are also explored by introducing a cut along the additional third strand and simulating the system relaxing to equilibrium. The analytical approach is based on the elastic rod model of a chain with bend and twist rigidities, while numerical simulation are performed on a coarse-grained model of three stranded chains whose stochastic dynamics is integrated using LAMMPS code. The study of such new configurations highlights the presence of a buckling transition similar to the one found with two strands, in which the coexistence of plectonemic and non formations is more pronounced than what previously observed. For such phase transition the geometric properties of the system influence directly the critical points positioning

Simulation of topological field theories by quantum computers

Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. These are called topological quantum filed theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm

Computing Embedded Contact Homology in Morse-Bott Settings

Given a contact three manifold $Y$ with a nondegenerate contact form $\lambda$, and an almost complex structure $J$ compatible with $\lambda$, its embedded contact homology $ECH(Y,\lambda)$ is defined (arXiv:1303.5789) and only depends on the contact structure. In this paper we explain how to compute ECH for Morse-Bott contact forms whose Reeb orbits appear in $S^1$ families, assuming the almost complex structure $J$ can be chosen to satisfy certain transversality conditions (this is the case for instance for boundaries of concave or convex toric domains, or if all the curves of ECH index one have genus zero). We define the ECH chain complex for a Morse-Bott contact form via an enumeration of ECH index one cascades. We prove using gluing results from arXiv:2206.04334 that this chain complex computes the ECH of the contact manifold. This paper and arXiv:2206.04334 fill in some technical foundations for previous calculations in the literature (arXiv:1608.07988, arXiv:math/0410061).Comment: 52 pages, comments welcom

A Class of P,TP,T-Invariant Topological Phases of Interacting Electrons

We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in 2+1-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. $P$ and $T$ invariance are maintained by a `doubling' of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-$T_c$ superconductivity. One is the `doubled' version (i.e. two opposite-chirality copies) of the U(1) chiral spin liquid. The second example corresponds to $Z_2$ gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly-organized discrete family. The physical inference is that a material manifesting the $Z_2$ gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly-constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.Comment: Typos fixed, references adde