22 research outputs found
The dual braid monoid
We construct a new monoid structure for Artin groups associated with finite
Coxeter systems. This monoid shares with the classical positive braid monoid a
crucial algebraic property: it is a Garside monoid. The analogy with the
classical construction indicates there is a ``dual'' way of studying Coxeter
systems, where the pair (W,S) is replaced by (W,T), with T the set of all
reflections. In the type A case, we recover the monoid constructed by
Birman-Ko-LeeComment: 42 pages. Major revision, many new result
Braid groups of imprimitive complex reflection groups
We obtain new presentations for the imprimitive complex reflection groups of
type and their braid groups for . Diagrams
for these presentations are proposed. The presentations have much in common
with Coxeter presentations of real reflection groups. They are positive and
homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms
correspond to group automorphisms. The new presentation shows how the braid
group is a semidirect product of the braid group of affine type
and an infinite cyclic group. Elements of are
visualized as geometric braids on strings whose first string is pure and
whose winding number is a multiple of . We classify periodic elements, and
show that the roots are unique up to conjugacy and that the braid group
is strongly translation discrete.Comment: published versio
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
Finite Gr\"obner--Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids
This paper shows that every Plactic algebra of finite rank admits a finite
Gr\"obner--Shirshov basis. The result is proved by using the combinatorial
properties of Young tableaux to construct a finite complete rewriting system
for the corresponding Plactic monoid, which also yields the corollaries that
Plactic monoids of finite rank have finite derivation type and satisfy the
homological finiteness properties left and right . Also, answering a
question of Zelmanov, we apply this rewriting system and other techniques to
show that Plactic monoids of finite rank are biautomatic.Comment: 16 pages; 3 figures. Minor revision: typos fixed; figures redrawn;
references update
Right-angled Artin groups and the cohomology basis graph
Let be a finite graph and let be the corresponding
right-angled Artin group. From an arbitrary basis of
over an arbitrary field, we construct a natural
graph from the cup product, called the \emph{cohomology
basis graph}. We show that always contains as a
subgraph. This provides an effective way to reconstruct the defining graph
from the cohomology of , to characterize the planarity of
the defining graph from the algebra of , and to recover many other
natural graph-theoretic invariants. We also investigate the behavior of the
cohomology basis graph under passage to elementary subminors, and show that it
is not well-behaved under edge contraction.Comment: 17 page
Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2
The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types An, Bn, Cn, Dn, and G2 (corresponding to the q-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right FP∞. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time