A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Random walks on braid groups are transient, with specific closure properties for certain braids.
We consider a random link, which is defined as the closure of a braid obtained from a random walk on the braid group. For such a random link, the expected value for the number of components was calculated by Jiming Ma. In this paper, we determine the most expected number of components for a random link, and further, co…
The study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.
A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.
Random braids that are formed by multiplying randomly chosen permutation braids are studied by analyzing their behavior under Garside's weighted decomposition and cycling. Using this analysis, we propose a polynomial-time algorithm to the conjugacy problem that is successful for random braids in overwhelming probabilit…
Braid theory optimizes neural network structures.
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list.…
We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to …
The reduced Burau representation of the braid group is obtained from the action of on the homology of an infinite cyclic cover of the disc with punctures. The group homology of braid groups with coefficients in the complexified reduced Burau representation is calculated. Our topolog…
The study proves super-rigidity of Gromov's random monster group for various types of groups.
New invariant for square-free integers derived from kei theory.
The UNKNOT problem solved using natural language processing and machine learning.
Polynomial invariant of quandles counts random link colorings.
Classic braids embed in virtual braids.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
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.
Formula found for braid index of -bridge braids.
Most simple braids have positive topological entropy.
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…
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.
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.
Magic braids can be made with leatherworking technique.
Study virtual braid groups, proving a key subgroup result.
Machine learning classifies braids and discovers new invariants.
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.
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.
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…
Polynomials with distinct critical values have braid monodromy groups equal to braid groups.
Invariants defined for braid systems under Hurwitz equivalence.
Paper defines generalized braids and proves their subgroup status.
Multi-agent Q-learning untangles braids, improving over training.
New family of braided Thompson groups introduced using recursive braids.
Paper explores relations between braid groups and their quotients.
The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in…
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…
A method to convert pretzel links into braids.
Paper proves homotopy braid group properties over integers and three strands.
Paper proves any twisted link can be described as a unique twisted braid.
New operations transform braids into P-fibered braids.
We give formulae for the first homology of the -braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the -braid group over the graph is torsion-free and the conjectures about the first h…
Paper defines a new braid invariant and shows it's commutative.
Positive braid knots have simple knot Floer homology.
We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…