467 research outputs found
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.Comment: 12 pages, 7 figure
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.Comment: 17 pages, AMSTeX, 17 figure
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.Comment: 11 pages, AMSTeX, 3 figure
Three-page embeddings of singular knots
Construction of a semigroup with 15 generators and 84 relations is given. The
center of this semigroup is in one-to-one correspondence with the set of all
isotopy classes of non-oriented singular knots (links with finitely many double
intersections in general position) in three-dimensional space.Comment: 14 pages, 5 figure
On the inverse braid monoid
Inverse braid monoid describes a structure on braids where the number of
strings is not fixed. So, some strings of initial may be deleted. In the
paper we show that many properties and objects based on braid groups may be
extended to the inverse braid monoids. Namely we prove an inclusion into a
monoid of partial monomorphisms of a free group. This gives a solution of the
word problem. Another solution is obtained by an approach similar to that of
Garside. We give also the analogues of Artin presentation with two generators
and Sergiescu graph-presentations.Comment: 18 pages, 5 figure
About presentations of braid groups and their generalizations
In the paper we give a survey of rather new notions and results which
generalize classical ones in the theory of braids. Among such notions are
various inverse monoids of partial braids. We also observe presentations
different from standard Artin presentation for generalizations of braids.
Namely, we consider presentations with small number of generators, Sergiescu
graph-presentations and Birman-Ko-Lee presentation. The work of V.V.Chaynikov
on the word and conjugacy problems for the singular braid monoid in
Birman-Ko-Lee generators is described as well.Comment: 35 pages, 18 figure
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 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.Comment: 13 pages, 2 figure
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.Comment: 5 page
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.Comment: 7 page
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 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.Comment: 16 page
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