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Garside calculus is the common mechanism that underlies a certain type of normal form for the elements of a monoid, a group, or a category. Originating from Garside's approach to Artin's braid groups, it has been extended to more and more general contexts, the latest one being that of categories and what are called Garside families. One of the benefits of this theory is to lead to algorithms solving effectively the naturally occurring problems, typically the Word Problem. The aim of this pape...
We study the Morton-Franks-Williams inequality for closures of simple
braids (also known as positive permutation braids). This allows to prove,
in a simple way, that the set of simple braids is a orthonormal basis for
the inner product of the Hecke algebra of the braid group defined by
K´alm´an, who first obtained this result by using an interesting connection
with Contact Topology. We also introduce a new technique to study the Homflypt polynomial for closures of positive brai...
An embedding of the m-times punctured disc into the n-times punctured
disc, for n > m, yields an embedding of the braid group on m strands Bm into the braid group on n strands Bn, called a geometric embedding. The main example consists of adding n−m trivial strands to the right of each braid on m strands. We show that geometric embeddings do not merge conjugacy classes, meaning that if the images of two elements in Bm by a geometric embedding are conjugate in Bn, the original elements are ...
We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type $A_{n-1}$ with respect to the simple elements (permutation braids) as generators. Instead of matrices of size $2^{n-1}\times 2^{n-1}$, we use matrices of size $p(n)\times p(n)$, where $p(n)$ is the number of partitions of $n$.
Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class group of a surface of genus $g$ with $n$ boundary components. We prove that ${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up to conjugation, a unique subgroup of index $2^{g-1}(2^{g}+1)$ up to conjugation, and the other proper subgroups of ${\mathcal{M}}_{g,n}$ are of index greater than $2^{g-1}(2^{g}+1)$. In particular, the minimum index for a proper subgroup of ${\mathcal{M}}_{g,n}$ ...
Analysing statistical properties of the normal forms of random braids, we observe that, except for an initial and a final region whose lengths are uniformly bounded (that is, the bound is independent of the length of the braid), the distributions of the factors of the normal form of sufficiently long random braids depend neither on the position in the normal form nor on the lengths of the random braids. Moreover, when multiplying a braid on the right, the expected number of factors in its nor...
We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of s...
We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irreducible Artin monoids of spherical type, extending a result by Caruso. Our results establish that the expected value of the penetration distance $pd(x,y)$ in an irreducible Art...
We present an algorithm to generate positive braids of a given length as words in Artin generators with a uniform probability. The complexity of this algorithm is polynomial in the number of strands and in the length of the generated braids. As a byproduct, we describe a finite state automaton accepting the language of lexicographically minimal representatives of positive braids that has the minimal possible number of states, and we prove that its number of states is exponential in the number...
A monoid $K$ is the internal Zappa-Sz\'ep product of two submonoids, if every element of $K$ admits a unique factorisation as the product of one element of each of the submonoids in a given order. This definition yields actions of the submonoids on each other, which we show to be structure preserving. We prove that $K$ is a Garside monoid if and only if both of the submonoids are Garside monoids. In this case, these factors are parabolic submonoids of $K$ and the Garside structure of $K$ can ...
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