On the generating function and growth of the positive singular braid monoid
We prove that the generating function of the positive singular braid monoid is rational and we give the exact formula for such monoid on three strands.
math.GR↗
arXiv subjects
V. V. Vershinin
Publications and source records attributed to V. V. Vershinin.
On homotopy braids
Homotopy braid group is the subject of the paper. First, linearity of homotopy braid group over the integers is proved. Then we prove that the group homotopy braid group on three strands is torsion free.
math.GR↗
Twisted Simplicial Groups and Twisted Homology of Categories
Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ δ\colon V(A)\longrightarrow \operatorname{End}(G), \quad v\mapsto δ_v $$ such that $δ_v\circ δ_w=δ_w\circ δ_v$ if there exists an edge in $A$ joining $v$ with $w$ or an arrow either from $v$ to $w$ or from $w$ to $v$. We give a canonical construction of twisted simplicial group as well as twisted homology for $A$ with a given twisted structure. Also we determine the homotopy type of of this simplicial group as the loop space over certain twisted smash product.
math.AT↗
Brunnian Braids and Lie Algebras
Brunnian braids have interesting relations with homotopy groups of spheres. In this work, we study the graded Lie algebra of the descending central series related to Brunnian subgroup of the pure braid group. A presentation of this Lie algebra is obtained.
math.GR↗
On Vassiliev invariants of braid groups of the sphere
We construct a universal Vassiliev invariant for braid groups of the sphere and the mapping class groups of the sphere with $n$ punctures. The case of a sphere is different from the classical braid groups or braids of oriented surfaces of genus strictly greater than zero, since Vassiliev invariants in a group without 2-torsion do not distinguish elements of braid group of a sphere.
math.GR↗
Brunnian Braids on Surfaces
We determine a set of generators for the Brunnian braids on a general surface $M$ for $M\not=S^2$ or $\RP^2$. For the case $M=S^2$ or $\RP^2$, a set of generators for the Brunnian braids on $M$ is given by our generating set together with the homotopy groups of a 2-sphere.
math.GT↗
On the pure virtual braid group $PV_3$
In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands. Moreover we prove that the presentation of PV_3 is aspherical. Finally we determine the cohomology ring and the associated graded Lie algebra of PV_3.
math.GT↗
On the inverse braid and reflection monoids of type $B$
There are well known relations between braid groups and symmetric groups, between Artin-Briskorn braid groups and Coxeter groups. Inverse braid monoid the same way is related to the inverse symmetric monoid. In the paper we show that similar relations exist between the inverse braid monoid of type $B$ and the inverse reflection monoid of type $B$. This gives a presentation of the last monoid.
math.GR↗
On the Lie algebras of surface pure braid groups
We consider the Lie algebra associated with the descending central series filtration of the pure braid group of a closed surface of arbitrary genus. R. Bezrukavnikov gave a presentation of this Lie algebra over the rational numbers. We show that his presentation remains true for this Lie algebra itself, i.e. over integers.
math.AT↗
On the inverse mapping class monoids
Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given $n$ punctures several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn-Nilsen-Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the inverse mapping class monoids. We also give an analogues of Artin presentation with two generators.
math.AT↗
On the Lie algebras associated with pure mapping class groups
Pure braid groups and pure mapping class groups of a punctured sphere have many features in common. In the paper the graded Lie algebra of the descending central series of the pure mapping class of a sphere is studied. A simple presentation of this Lie algebra is obtained.
math.AT↗
Braids, their properties and generalizations
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure braid group and the Markov normal form are given next. Garside normal form and his solution of the conjugacy problem are presented as well as more recent results on the ordering and on the linearity of braid groups. Next topics are the generalizations of braids, their homological properties and connections with the other mathematical fields, like knot theory (via Alexander and Markov theorems) and homotopy groups of spheres.
math.GR↗
On the singular braid monoid
Garside's results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.
math.GR↗
On Homology of Virtual Braids and Burau Representation
Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. Virtual braids correspond naturally to virtual knots. We consider the group of virtual braids on n strings VB_n and its Burau representation, in particular we study their homological properties. We prove that the plus-construction of the classifying space of the virtual braid group on the infinite number of strings is an infinite loop space which is equivalent to a product of Q(S^0), S^1 and an infinite loop space Y. Connections with the K-functor of the integers are discussed.
math.GT↗
On Vassiliev Invariants for Links in Handlebodies
The notion of Vassiliev algebra in case of hanlebodies is developed. The analogues of the results of John Baez for links in handlebodies are proved. That means that there exists 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.
q-alg↗