Constructing unlabelled lattices

We present an improved orderly algorithm for constructing all unlabelled lattices up to a given size, that is, an algorithm that constructs the minimal element of each isomorphism class relative to some total order. Our algorithm employs a stabiliser chain approach for cutting branches of the search space that cannot contain a minimal lattice; to make this work, we grow lattices by adding a new layer at a time, as opposed to adding one new element at a time, and we use a total order that is compatible with this modified strategy. The gain in speed is between one and two orders of magnitude. As an application, we compute the number of unlabelled lattices on 20 elements

Algorithms for Garside calculus

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 paper is to present and solve these algorithmic questions in the new extended framework

Normal forms of random braids

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 normal form that are modified, called the expected penetration distance, is uniformly bounded. We explain these observations by analysing the growth rates of two regular languages associated to normal forms of elements of Garside groups, respectively to the modification of a normal form by right multiplication. A universal bound on the expected penetration distance in a Garside group yields in particular an algorithm for computing normal forms that has linear expected running time

On the penetration distance in Garside monoids

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 Artin monoid of spherical type is bounded independently of the length of x, if xis chosen uniformly among all elements of given canonical length and y is chosen uniformly among all atoms; the latter in particular explains observations made by Thurston in the context of the braid group, and it shows that all irreducible Artin monoids of spherical type exhibit an analogous behaviour. Our results also give an affirmative answer to a question posed by Dehornoy

Bacterial genomics and computational group theory : the BioGAP package for GAP

Bacterial genomes can be modelled as permutations of conserved regions. These regions are sequences of nucleotides that are identified for a set of bacterial genomes through sequence alignment, and are presumed to be preserved through the underlying process, whether through chance or selection. Once a correspondence is established between genomes and permutations, the problem of determining the evolutionary distance between genomes (in order to construct phylogenetic trees) can be tackled by use of group-theoretical tools. Here we review some of the resulting problems in computational group theory and describe BioGAP, a computer algebra package for genome rearrangement calculations, implemented in GAP

Bacterial phylogeny in the Cayley graph

Many models of genome rearrangement involve operations that are self-inverse, and hence generate a group acting on the space of genomes. This gives a correspondence between genome arrangements and the elements of a group, and consequently, between evolutionary paths and walks on the Cayley graph. Many common methods for phylogenetic reconstruction rely on calculating the minimal distance between two genomes; this omits much of the other information available from the Cayley graph. In this paper, we begin an exploration of some of this additional information, in particular describing the phylogeny as a Steiner tree within the Cayley graph, and exploring the "interval" between two genomes. While motivated by problems in systematic biology, many of these ideas are of independent group-theoretic interest