The group Gn2 and invariants of free links valued in free groups

2015 ◽  
Vol 24 (13) ◽  
pp. 1541010 ◽  
Author(s):  
V. O. Manturov ◽  
S. Kim
Keyword(s):  
Dynamical Systems ◽  
Free Product ◽  
Equivalence Relation ◽  
Braid Group ◽  
Knot Theory ◽  
Free Groups ◽  
Group Element ◽  
Group Presentation ◽  
End Points ◽  
And Topology

In this paper, we define an invariant of free links valued in a free product of some copies of [Formula: see text]. In [Non-Reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208v1] the second author constructed a connection between classical braid group and group presentation generated by elements corresponding to horizontal trisecants. This approach does not apply to links nor tangles because it requires that when counting trisecants, we have the same number of points at each level. For general tangles, trisecants passing through one component twice may occur. Free links can be obtained from tangles by attaching two end points of each component. We shall construct an invariant of free links and free tangles valued in groups as follows: we associate elements in the groups with 4-valent vertices of free tangles (or free links). For a free link with enumerated component, we “read” all the intersections when traversing a given component and write them as a group element. The problem of “pure crossings” of a component with itself by using the following statement: if two diagrams with no pure crossings are equivalent then they are equivalent by a sequence of moves where no intermediate diagram has a pure crossing. This statement is a result of a sort that an equivalence relation within a subset coincides with the equivalence relation induced from a larger set and it is interesting by itself.

On braids and groups Gnk

2015 ◽  
Vol 24 (13) ◽  
pp. 1541009 ◽  
Author(s):  
Vassily Olegovich Manturov ◽  
Igor Mikhailovich Nikonov
Keyword(s):  
Dynamical Systems ◽  
Braid Group ◽  
Knot Theory ◽  
Free Groups ◽  
Braid Groups ◽  
The Other ◽  
Pure Braid Group ◽  
And Topology ◽  
Other Hand ◽  
Definition Of

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.

scholarly journals On groups Gnk, braids and Brunnian braids

2016 ◽  
Vol 25 (13) ◽  
pp. 1650078
Author(s):  
S. Kim ◽  
V. O. Manturov
Keyword(s):  
Dynamical Systems ◽  
Braid Group ◽  
General Position ◽  
Knot Theory ◽  
The Other ◽  
Geometric Meaning ◽  
Free Products ◽  
Other Hand ◽  
Abstract Structure

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.

scholarly journals The absorption theorem for affable equivalence relations

2008 ◽  
Vol 28 (5) ◽  
pp. 1509-1531 ◽  
Author(s):  
THIERRY GIORDANO ◽  
HIROKI MATUI ◽  
IAN F. PUTNAM ◽  
CHRISTIAN F. SKAU
Keyword(s):  
Dynamical Systems ◽  
Equivalence Relation ◽  
Closed Subset ◽  
Cantor Set ◽  
Equivalence Relations ◽  
Orbit Structure ◽  
Orbit Equivalent ◽  
Finite Set ◽  
Significant Generalization

AbstractWe prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal ℤn-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys.24 (2004), 441–475] . However, we shall need a few key results from the above paper in order to prove the absorption theorem.

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

A GENERALIZED REDUCTION PROCEDURE FOR DYNAMICAL SYSTEMS

1991 ◽  
Vol 06 (37) ◽  
pp. 3445-3453 ◽  
Author(s):  
G. LANDI ◽  
G. MARMO ◽  
G. SPARANO ◽  
G. VILASI
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Equivalence Relation ◽  
Reduction Procedure

We describe a reduction procedure for dynamical systems. If Γ is a dynamical vector field on a manifold M, a reduced system is obtained by projecting Γ to a manifold Σ/≈ where Σ is a submanifold of M invariant under Γ and ≈ is a suitable equivalence relation.

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals Dynamical systems and topology optimization

2010 ◽  
Vol 42 (2) ◽  
pp. 179-192 ◽  
Author(s):  
Anders Klarbring ◽  
Bo Torstenfelt
Keyword(s):  
Topology Optimization ◽  
Dynamical Systems ◽  
And Topology

scholarly journals Virtual braids and virtual curve diagrams

2015 ◽  
Vol 24 (13) ◽  
pp. 1541001 ◽  
Author(s):  
Oleg Chterental
Keyword(s):  
Automorphism Group ◽  
Braid Group ◽  
Knot Theory ◽  
Class Group ◽  
Mapping Class ◽  
Virtual Knot ◽  
Injective Homomorphism ◽  
Virtual Links ◽  
Upper Half Plane ◽  
Virtual Knot Theory

There is a well-known injective homomorphism [Formula: see text] from the classical braid group [Formula: see text] into the automorphism group of the free group [Formula: see text], first described by Artin [Theory of Braids, Ann. Math. (2) 48(1) (1947) 101–126]. This homomorphism induces an action of [Formula: see text] on [Formula: see text] that can be recovered by considering the braid group as the mapping class group of [Formula: see text] (an upper half plane with [Formula: see text] punctures) acting naturally on the fundamental group of [Formula: see text]. Kauffman introduced virtual links [Virtual knot theory, European J. Combin. 20 (1999) 663–691] as an extension of the classical notion of a link in [Formula: see text]. There is a corresponding notion of a virtual braid, and the set of virtual braids on [Formula: see text] strands forms a group [Formula: see text]. In this paper, we will generalize the Artin action to virtual braids. We will define a set, [Formula: see text], of “virtual curve diagrams” and define an action of [Formula: see text] on [Formula: see text]. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in [Formula: see text]. In the papers [V. G. Bardakov, Virtual and welded links and their invariants, Siberian Electron. Math. Rep. 21 (2005) 196–199; V. O. Manturov, On recognition of virtual braids, Zap. Nauch. Sem. POMI 299 (2003) 267–286], an extension [Formula: see text] of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that [Formula: see text] is not injective by exhibiting a non-trivial virtual braid in the kernel when [Formula: see text].

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