scholarly journals On parabolic submonoids of a class of singular Artin monoids

2007 ◽  
Vol 82 (1) ◽  
pp. 29-37
Author(s):  
Noelle Antony
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Artin Groups ◽  
Parabolic Subgroups ◽  
Vassiliev Invariants ◽  
Sole Purpose

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

scholarly journals PALINDROMES AND ORDERINGS IN ARTIN GROUPS

2010 ◽  
Vol 19 (02) ◽  
pp. 145-162 ◽  
Author(s):  
FLORIAN DELOUP
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Type A ◽  
Reverse Order ◽  
Artin Groups ◽  
Left Order ◽  
Group B ◽  
Half Twist

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn →Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

Knot Theory and Statistical Mechanics

1997 ◽  
Vol 11 (01n02) ◽  
pp. 39-49 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Statistical Mechanics ◽  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Scattering Amplitude ◽  
Basic Structure ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Quantum Link

This paper gives a self-contained exposition of the basic structure of quantum link invariants as state summations for a vacuum-vacuum scattering amplitude. Models of Vaughan Jones are expressed in this context. A simple proof is given that an important subset of these invariants are built from Vassiliev invariants of finite type.

scholarly journals Free groups and finite-type invariants of pure braids

2002 ◽  
Vol 132 (1) ◽  
pp. 117-130 ◽  
Author(s):  
JACOB MOSTOVOY ◽  
SIMON WILLERTON
Keyword(s):  
Finite Type ◽  
Free Group ◽  
Free Groups ◽  
Braid Groups ◽  
Point Of View ◽  
Magnus Expansion ◽  
Vassiliev Invariants ◽  
Integer Coefficients ◽  
Finite Type Invariants ◽  
Theoretic Point

In this paper finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of each type for all pure braid groups.

scholarly journals ON VASSILIEV INVARIANTS FOR LINKS IN HANDLEBODIES

1998 ◽  
Vol 07 (05) ◽  
pp. 701-712 ◽  
Author(s):  
VLADIMIR V. VERSHININ
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Special Class ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Relative Version ◽  
One To One

The notion of Vassiliev algebra in case of handlebodies is developed. Analogues of the results of J. Baez for links in handlebodies are proved. This implies that there is a one-to-one correspondence between the special class of finite type invariants of links in hanlebodies and the homogeneous Markov traces on Vassiliev algebras. This approach uses the singular braid monoid and braid group in a handlebody and the generalizations of the theorem of J. Alexander and the theorem of A. A. Markov for singular links and braids and the relative version of Markov's theorem.

On marked braid groups

2015 ◽  
Vol 24 (13) ◽  
pp. 1541005 ◽  
Author(s):  
Denis A. Fedoseev ◽  
Vassily O. Manturov ◽  
Zhiyun Cheng
Keyword(s):  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Arbitrary Group ◽  
Natural Group ◽  
Additional Information ◽  
Reidemeister Moves ◽  
Braid Theory ◽  
Important Notion

In this paper, we introduce [Formula: see text]-braids and, more generally, [Formula: see text]-braids for an arbitrary group [Formula: see text]. They form a natural group-theoretic counterpart of [Formula: see text]-knots, see [V. O. Manturov; Reidemeister moves and groups, preprint (2014), arXiv:1412.8691]. The underlying idea used in the construction of these objects — decoration of crossings with some additional information — generalizes an important notion of parity introduced by the second author (see [V. O. Manturov, Parity in knot theory, Sb. Math. 201(5) (2010) 693–733]) to different combinatorically geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems.

A GEOMETRIC AND ALGEBRAIC DESCRIPTION OF ANNULAR BRAID GROUPS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 85-97 ◽  
Author(s):  
RICHARD P. KENT ◽  
DAVID PEIFER
Keyword(s):  
Finite Type ◽  
Cyclic Group ◽  
Semidirect Product ◽  
Braid Group ◽  
Braid Groups ◽  
Artin Group ◽  
Infinite Cyclic Group ◽  
Coxeter Graph ◽  
Infinite Type ◽  
Algebraic Description

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn - 1. This provides a new example of an infinite type Artin group which injects into a finite type Artin group. In fact, we show that the affine braid group with Coxeter graph Ãn - 1 injects into the braid group on n + 1 stings. Recently it has been shown that the braid groups are linear, see [3]. Therefore, this shows that the affine braid groups are also linear.

RANDOMIZED METHODS IN ARTIN GROUPS OF FINITE TYPE

2007 ◽  
Vol 17 (04) ◽  
pp. 859-868
Author(s):  
ROBERT W. BRADSHAW ◽  
STEPHEN P. HUMPHRIES
Keyword(s):  
Finite Type ◽  
Braid Groups ◽  
Artin Groups ◽  
Randomized Methods

We investigate the use of random algorithms for the solution of the conjugacy problems for certain types of Artin groups of finite type including the braid groups.

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 Elementary equivalence in Artin groups of finite type

2018 ◽  
Vol 28 (02) ◽  
pp. 331-344
Author(s):  
Arpan Kabiraj ◽  
T. V. H. Prathamesh ◽  
Rishi Vyas
Keyword(s):  
Finite Type ◽  
Braid Groups ◽  
Mapping Class ◽  
Artin Groups ◽  
Elementary Equivalence ◽  
Closed Surfaces ◽  
Text Fragment ◽  
The Relationship ◽  
Geometric Analogue ◽  
Elementary Theories

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. The [Formula: see text] fragment suffices to distinguish the elementary theories of the groups in question. As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces — a geometric analogue of braid groups — are elementarily equivalent if and only if they are isomorphic.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

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