Statistical and probabilistic characteristics of locally free group with growing number of generators are defined and their application to statistics of braid groups is given.
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48 results for “braiding statistics”
Study on Alexander polynomials in braids, linking number theory and topology.
problem Distribution of Alexander polynomials in specific families of braids.
method Exploration of arithmetic invariants and analogies with number theory.
result New directions in arithmetic topology and statistics.
After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point parti…
Quantum Statistics and Spacetime Surgerycond-mat.str-el
We apply the geometric-topology surgery theory on spacetime manifolds to study the constraints of quantum statistics data in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable creating anyon excitations of particles and strings, well-defined in gapped st…
Quantum surgery formulas link statistics and topology.
problem Formulate constraints for quantum statistics of entangled systems.
method Geometric-topology surgery theory on spacetime manifolds.
result New quantum surgery formulas and constraints derived.
Study of circle configurations in the plane, proving aspherical space and computing fundamental groups.
problem Understanding the space of configurations of circles in the plane.
method Proved the space is aspherical and computed fundamental groups of its components.
result Fundamental groups are iterated semidirect products of braid groups, with structure dictated by a finite rooted tree.
Universal Braiding Quantum Gatesquant-ph
Simple construction for universal quantum gates.
problem Designing efficient quantum gates for topological computers.
method Demonstrated a simple construction for unitary solutions of the braided Yang-Baxter equation in any dimension.
result Proved the existence of universal quantum gates in any dimension.
Classic braids embed in virtual braids.
problem Embedding classic braids in virtual braids.
method Elementary proof using generalizations of braid groups.
result Classical braid group injects into virtual braid group.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
Study on groups of unrestricted virtual braids and their properties.
problem Understanding the structure and properties of virtual braids and links.
method Exploration of unrestricted virtual braids, fused links, flat virtual braids, and virtual Gauss braids.
result Definition and study of groups of flat virtual braids and virtual Gauss braids, including their linearity.
A simple and elegant arrangement of stock components of a portfolio (market index-DJIA) in a recent paper [1], has led to the construction of crossing of stocks diagram. The crossing stocks method revealed hidden remarkable algebraic and geometrical aspects of stock market. The present paper continues to uncover new ma…
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
Satellite links of fully positive braids are characterized.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
Formula found for braid index of -bridge braids.
problem Finding a formula for the braid index of -bridge braids.
method Elementary, effective, self-contained proof.
result Closed form formula for braid index of -bridge braids.
Most simple braids have positive topological entropy.
problem Understanding the topological entropy of simple braids.
method Reduction from simple braids to non-simple 3-strand braids.
result The proportion of simple braids with positive entropy approaches 100% as the number of strands increases.
This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close contr…
Paper proves Markov's theorem for extended welded braids and links.
problem Generalizing welded links to extended welded links and braids.
method Following Kamada's approach, proving Alexander and Markov's theorems for extended welded braids and links.
result Proves versions of Alexander and Markov's theorems for extended welded braids and links.
We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid wo…
The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective…
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensionscond-mat.str-el
Discovering topological quantum field theories in 2+1 and 3+1 dimensions.
problem Exploring topological orders in condensed matter lattice models.
method Calculating braiding statistics and link invariants of anyon excitations.
result Identifying new spin topological quantum field theories with specific knot/link invariants.
Polynomial-time algorithm for virtual braid triviality.
problem Determining if virtual braid words represent the trivial braid.
method An -time algorithm for .
result Determines triviality of virtual braid words efficiently.
This paper identifies braided 3-belts that can be written in a braid-only form.
problem Identifying braided 3-belts that can be written in a braid-only form.
method Developed an algorithm to calculate the braid word for braided 3-belts and determined the conditions for knotted boundaries.
result Identified the set of braided 3-belts that can be written in a braid-only form and derived a formula for the Jones polynomial for knotted boundaries.
Magic braids can be made with leatherworking technique.
problem Determine which braids can be made with leatherworking technique.
method Explore their relation to several braid groups.
result Magic braids can be made with leatherworking technique.
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
Machine learning classifies braids and discovers new invariants.
problem Classifying and discovering invariants of braids and flat braids.
method Supervised learning with neural networks to classify braids as trivial or non-trivial.
result Found new convenient invariants of braids, including a complete invariant of flat braids.
Virtual Braidsmath.GT
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…
This paper extends braid lifting to coloured braid groupoids for all simple disc covers.
problem Lifting braids to homeomorphisms on branched covers of the disc.
method Defines a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc.
result Characterizes the lift of every coloured braid, recovering classical lifting on liftable braids.
Paper studies special braids from chromatic configuration spaces.
problem Understanding braids with crossing-changeable properties.
method Theory of chromatic configuration spaces and braids.
result Introduced braids with the ability to intersect and untangle.
New virtual braid group structure and presentation found.
problem Understanding the structure of virtual pure braid groups.
method Cabling construction and HNN-extension approach.
result New presentation of and groups.
Boundary Braidsmath.GR
New concept of boundary braids defined for disk configurations.
problem Defining and studying braids with points fixed on the boundary.
method Using configuration spaces and fundamental groups, defining boundary braids and analyzing their geometric properties.
result Boundary braids form a subcomplex that metrically splits into a Euclidean polyhedron and a smaller rank dual braid complex.
The notion of a braid is generalized into two and three dimensions. Two-dimensional braids are described by braid monodromies or graphics called charts. In this paper we introduce the notion of curtains, and show that three-dimensional braids are described by braid monodromies or curtains.
New braid representations using virtual knot theory.
problem No classical features in virtual knot theory.
method Construct new braid representations using virtual knot theory.
result New representations of classical braids.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…
Classifies closed surface braids up to isotopy and homeomorphism.
problem Classifying closed surface braids up to isotopy and homeomorphism.
method Algebraic classification using surface braid group and mapping class group actions.
result Braids close to isotopic or homeomorphic links if and only if they are conjugate or in the same orbit of the outer action of the mapping class group.
Invariants defined for braid systems under Hurwitz equivalence.
problem Invariants for braid systems under Hurwitz equivalence.
method Crossing matrix and polynomial invariant introduced.
result Invariant defined for surface braids and surface links.
Paper defines generalized braids and proves their subgroup status.
problem Understanding the structure of generalized braids and knots.
method Defined generalized braid theories and computed their generating sets.
result Quasitoric normal generalized braids form a subgroup of normal generalized braid group.
Polynomials with distinct critical values have braid monodromy groups equal to braid groups.
problem Understanding the structure of braid monodromy groups of polynomials.
method Analyzing the critical values of polynomials to determine their braid monodromy groups.
result The braid monodromy group of a polynomial equals the braid group if the polynomial has distinct critical values.
Braids with more twists than strands achieve their braid index.
problem Understanding the braid index of closures of braids with fractional Dehn twist coefficients.
method Characterizing fractional Dehn twist coefficients in terms of the Upsilon function and proving the braid index via homogenization of knot concordance homomorphisms.
result Proves that -braids with fractional Dehn twist coefficient larger than realize the braid index of their closure.
Multi-agent Q-learning untangles braids, improving over training.
problem Untangling braids using reinforcement learning.
method Two competing players learn to tangle and untangle braids in the OpenAI Gym environment.
result The system improves at untangling braids with more training, producing good tangled examples.
New family of braided Thompson groups introduced using recursive braids.
problem Dehornoy-Brin braided Thompson group generalization.
method Using recursive braids and strand diagrams to define new groups.
result New groups are finitely generated if is finitely generated.
Multi-crossing Braidsmath.GT
Generalizes braid theory to multi-crossing braids.
problem Knots with more than two strands at a crossing.
method Extends Alexander's Theorem to multi-crossing braids.
result Every link can be represented in an -crossing braid form for any even .
The study refines contingency matrices for complex stratification and braid group cohomology.
problem Combinatorics of contingency matrices and their applications.
method Refinement of complex stratification and study of braid group cohomology.
result Totally positive meta-matrix formed by contingency matrix sizes.
The aim of the present note is to show that the natural map from classical braids to virtual braids is an inclusion; this proof does not use any complete invariants of classical braids; it is based on the projection from virutal braids to classical braids (similar to the one given in \cite{Projection}); this projection…
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
Twisted braidsmath.GT
Paper proves any twisted link can be described as a unique twisted braid.
problem Understanding twisted links and braids.
method Proved using Alexander and Markov Theorems for classical braids and links.
result Any twisted link can be described as the closure of a unique twisted braid.
Polynomial braid combingmath.GT
Polynomial algorithm solves word problem in braid groups on surfaces.
problem Word problem in braid groups on surfaces.
method Theory of straight line programs for polynomial braid combing.
result First algorithm to solve word problem in polynomial time for braid groups on surfaces.