Braid Invariants Research Articles

Machine Learning Discovers Invariants of Braids and Flat Braids

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our machine learning takes the form of supervised learning, specifically multilayer perceptron neural networks. When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new invariants of braids and prove several theorems related to them. This work evolves from our experiments exploring how different types of AI cope with untangling braids with 3 strands, this is why we concentrate mostly on braids with 3 strands.

Milnor invariants of braids and welded braids up to homotopy

Milnor invariants of braids and welded braids up to homotopy

Evaluations of annular Khovanov–Rozansky homology

We describe the universal target of annular Khovanov–Rozansky link homology functors as the homotopy category of a free symmetric monoidal linear category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov–Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.

Invariants of Classical Braids Valued In $$ {G}_n^2 $$

The aim of the present note is to enhance groups \( {G}_n^3 \) and to construct new invariants of classical braids. In particular, we construct invariants valued in \( {G}_N^2 \) groups. In groups \( {G}_n^2 \), the identity problem is solved; besides, their structure is much simpler than that of \( {G}_n^3 \).

Representations of Gn3 and related groups

In this paper, we study representations of [Formula: see text]-like groups. The group [Formula: see text] itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of [Formula: see text]-like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the [Formula: see text]-like group structure.

Transverse link invariants from the deformations of Khovanov 𝔰𝔩3–homology

In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $\beta_3$-invariants) which are transverse braid invariants. This family includes Wu's $\psi_{3}$-invariant. Furthermore, we analyse the vanishing of the homology classes of the $\beta_3$-invariants and relate it to the vanishing of Plamenevskaya's and Wu's invariants. Finally, we use the $\beta_3$-invariants to produce some Bennequin-type inequalities.

A Bennequin-Type Inequality and Combinatorial Bounds

In this paper, we provide a new Bennequin-type inequality for the Rasmussen–Beliakova–Wehrli invariant, featuring the numerical transverse braid invariants (the c-invariants) introduced by the author. From the Bennequin type-inequality and a combinatorial bound on the value of the c-invariants we deduce a new computable bound on the Rasmussen invariant.

Fundamental groups, 3-braids, and effective estimates of invariants

We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure 3-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the word representing the image of the braid in the braid group modulo its center. The bounds differ by a multiplicative constant not depending on the word. Respective bounds are given for all 3-braids. We also obtain effective upper and lower bounds for the entropy of pure 3-braids in these terms. The proof leads to the study of the extremal length (in the sense of Ahlfors) of classes of curves representing elements of the fundamental group of the twice punctured complex plane.

On groups Gnk, braids and Brunnian braids

In [V. O. Manturov, arXiv:1501.05208v1 ], the second author defined the [Formula: see text]-free braid group with [Formula: see text] strands [Formula: see text]. These groups appear naturally as groups describing dynamical systems of [Formula: see text] particles in some “general position”. Moreover, in [V. O. Manturov and I. M. Nikonov, J. Knot Theory Ramification 24 (2015) 1541009] the second author and Nikonov showed that [Formula: see text] is closely related to classical braids. The authors showed that there are homomorphisms from the pure braids group on [Formula: see text] strands to [Formula: see text] and [Formula: see text] and they defined homomorphisms from [Formula: see text] to the free products of [Formula: see text]. That is, there are invariants for pure free braids by [Formula: see text] and [Formula: see text]. On the other hand in [D. A. Fedoseev and V. O. Manturov, J. Knot Theory Ramification 24(13) (2015) 1541005, 12 pages] Fedoseev and the second author studied classical braids with addition structures: parity and points on each strands. The authors showed that the parity, which is an abstract structure, has geometric meaning — points on strands. In [S. Kim, arXiv:submit/1548032], the first author studied [Formula: see text] with parity and points. the author constructed a homomorphism from [Formula: see text] to the group [Formula: see text] with parity. In the present paper, we investigate the groups [Formula: see text] and extract new powerful invariants of classical braids from [Formula: see text]. In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids.

An elementary proof that classical braids embed in virtual braids

The purpose of this paper is to prove that the natural mapping of classical braids to virtual braids is an embedding. The proof does not use any complete invariants of classical braids; it is based on a projection from (colored) virtual braids onto classical braids (which is similar to the projection in [6]); this projection is the identity mapping on the set of classical braids. It is well defined do not only for the group of (colored) virtual braids but also for the quotient group of the group of (colored) virtual braids by the so-called virtualization motion. The idea of this projection is closely related to the notion of parity and the groups Gnk introduced by the author in [3].

On braids and groups Gnk

In [Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208.] the first author gave the definition of [Formula: see text]-free braid groups [Formula: see text]. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to [Formula: see text]-free braid groups for important cases [Formula: see text]. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.

Two- and three-cocycles for Laver tables

We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of finite structures obeying the left-self-distributivity law; in particular, we describe simple explicit bases. This provides a number of new positive braid invariants and paves the way for further potential topological applications. An important tool for constructing a combinatorially meaningful basis of 2-cocycles is the right-divisibility relation on Laver tables, which turns out to be a partial ordering.

Coloring link diagrams and Conway-type polynomial of braids

In this paper we define and present a simple combinatorial formula for a 3-variable Laurent polynomial invariant I(a,z,t) of conjugacy classes in Artin braid group Bm. We show that the Laurent polynomial I(a,z,t) satisfies the Conway skein relation and the coefficients of the 1-variable polynomial t−kI(a,z,t)|a=1,t=0 are Vassiliev invariants of braids.

Oscillatory solutions of fourth order conservative systems via the Conley index

In this paper we investigate periodic solutions of second order Lagrangian systems which oscillate around equilibrium points of center type. The main ingredients are the discretization of second order Lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed by Ghrist et al. (2003) [5]. The problem with applying this topological theory directly is that the braid types in our analysis are so-called improper. This implies that the braid invariants do not entirely depend on the topology: the relevant braid classes are non-isolating neighborhoods of the flow, so that their Conley index is not universal. In first part of this paper we develop the theory of the braid invariant for improper braid classes and in the second part this theory is applied to second order Lagrangian system and in particular to the Swift–Hohenberg equation.

Conway polynomial and Magnus expansion

The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We describe explicitly the resulting mapping from horizontal chord diagrams on 3 strands to univariate polynomials and evaluate it on the Drinfeld associator obtaining, conjecturally, a beautiful generating function whose coefficients are integer combinations of multiple zeta values.

The Order of Bifurcation Points in Fourth Order Conservative Systems via Braids

In second order Lagrangian systems bifurcation branches of periodic solutions preserve certain topological invariants. These invariants are based on the observation that periodic orbits of a second order Lagrangian lie on 3-dimensional (noncompact) energy manifolds and the periodic orbits may have various linking and knotting properties. The main ingredients defining the topological invariants are the discretization of second order Lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed in [R. W. Ghrist, J. B. Van den Berg, and R. C. Vandervorst, Invent. Math., 152 (2003), pp. 369–432]. In the first part of this paper we recall the essential theory of braid invariants, and in the second part this theory is applied to second order Lagrangian systems and in particular to the Swift–Hohenberg equation. We show that the invariants yield forcing relations on bifurcation branches. We quantify this principle via an order relation on the topological type of a bifurc...

Knot and braid invariants from contact homology II

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branched covers for general knots. In the appendix we show that the cord ring is determined by the fundamental group and peripheral structure of a knot and give applications.

Knot and braid invariants from contact homology I

We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative contact homology of certain Legendrian tori in five-dimensional contact manifolds. We present several computations and derive a relation between the knot invariant and the determinant.

Biquandles and virtual links

In this paper we define a birack and a biquandle, generalizing the notion of a rack and a quandle. This gives rise to natural invariants of virtual knots and braids. Some of the properties of biracks and biquandles are explained in this paper. Applications to particular virtual braids and links are given.

FINITE TYPE INVARIANTS FOR SINGULAR SURFACE BRAIDS ASSOCIATED WITH SIMPLE 1-HANDLE SURGERIES

S. Kamada introduced finite type invariants of knotted surfaces in 4-space associated with finger moves and 1-handle surgeries. In this paper, we define finite type invariants of surface braids associated with simple 1-handle surgeries and prove that a certain set of finite type invariants controls all finite type invariants. As a consequence, we see that every finite type invariant is not a complete invariant.