arXiv research

Invariants defined for braid systems under Hurwitz equivalence.

Let $D_n$ denote the $n$-punctured disk in the complex plane, where the punctures are on the real axis. An $n$-braid $α$ is said to be \emph{reducible} if there exists an essential curve system $\C$ in $D_n$, called a \emph{reduction system} of $α$, such that $α*\C=\C$ where $α*\C$ denotes the action of the braid $α$ o…

The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.

Introduces a reduction system for Artin-Tits groups, improving algorithms and proving periodicity results.

The paper defines biquandles for groups and constructs a homomorphism.

Multi-agent Q-learning untangles braids, improving over training.

In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators of the braid group.

In [V.O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, arxiv:1501.05208] the first named author gave the definition of $k$-free braid groups $G_n^k$. Here we establish connections between free braid groups, classical braid groups and free groups: we describe e…

Classifies orbits of Hurwitz actions on dihedral quandles.

Study extends symmetries of sphere points to surface mapping classes.

Study deformation spaces of irregular isomonodromy systems on Riemann surfaces.

For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of closed orientable surfaces, B type groups of Artin-Brieskorn, and allow us to study a…

We propose an algorithm for deciding whether a given braid is pseudo-Anosov, reducible, or periodic. The algorithm is based on Garside's weighted decomposition and is polynomial-time in the word-length of an input braid. Moreover, a reduction system of circles can be found completely if the input is a certain type of r…

New groups constructed from tree automorphisms, proving finiteness.

Unified Long-Moody and Katz methods for constructing local systems.

The comparison principle for scalar second order parabolic PDEs on functions $u(t,x)$ admits a topological interpretation: pairs of solutions, $u^1(t,\cdot)$ and $u^2(t,\cdot)$, evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $\{u^α(t,\cdot)\}_{α=1}^…

A new approach uses circuit topology to study complex polymer interactions.

Study braid group actions on exceptional sequences using branched coverings.

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…

Maximizes mixing efficiency in surface braids.

New method for computing Kauffman bracket skein module of lens spaces using unoriented braids.

We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conugacy problem given by the authors in a previous paper, are two main steps for solving conjugacy systems, thus breaking recently discovered crypto…

The paper introduces invariants to describe period-doubling routes to chaos in dynamical systems.

Researchers develop a new basis to simplify solving infinite systems for HOMFLYPT skein module of lens spaces.

In \cite{Manturov} the second author defined the $k$-free braid group with $n$ strands $G_{n}^{k}$. These groups appear naturally as groups describing dynamical systems of $n$ particles in some "general position". Moreover, in \cite{ManturovNikonov} the second author and I.M.Nikonov showed that $G_{n}^{k}$ is closely r…

Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to diagrams of the initial object. The main tool is the notion of {\em free $k$-braid …

We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy takin…

The Dynnikov coordinate system puts global coordinates on the boundary of Teichmüller space of an $n$--punctured disk. We survey the Dynnikov coordinate system, and investigate how we use this coordinate system to study pseudo--Anosov braids making use of results from Thurston's theory on surface homeomorphisms.

fkcompute calculates a knot invariant from a braid presentation.

In this paper we work toward the Homflypt skein module of the lens spaces $L(p,1)$, $\mathcal{S}(L(p,1))$, using braids. In particular, we establish the connection between $\mathcal{S}({\rm ST})$, the Homflypt skein module of the solid torus ST, and $\mathcal{S}(L(p,1))$ and arrive at an infinite system, whose solution…

Researchers compute the skein module of a solid torus using braids.

Study of Lorenz links and T-links, showing equivalence and unique presentations.

In this paper we present recent results toward the computation of the HOMFLYPT skein module of the lens spaces $L(p,1)$, $\mathcal{S}\left(L(p,1) \right)$, via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, $X$, for knots and links in ST, the universal analogue of th…

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.…

In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel knots cannot be unknotted using only positive crossing changes. We also describe …

Characterizes winding of braided vector fields in tubular domains.

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimension…

The paper computes the Kauffman bracket skein module of $S^1 imes S^2$ via braids.

Classic braids embed in virtual braids.

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces $L(p,1)$ from the HOMFLYPT skein module of the solid torus, $\mathcal{S}({\rm ST})$, it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant $X$ for knots and links in the solid torus, braid ba…

Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.

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 $n$-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 provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.