411 research outputs found
The group of parenthesized braids
We investigate a group that includes Artin's braid group
and Thompson's group . The elements of are
represented by braids diagrams in which the distances between the strands are
not uniform and, besides the usual crossing generators, new rescaling operators
shrink or strech the distances between the strands. We prove that
is a group of fractions, that it is orderable, admits a non-trivial
self-distributive structure, i.e., one involving the law ,
embeds in the mapping class group of a sphere with a Cantor set of punctures,
and that Artin's representation of into the automorphisms of a free
group extends to
Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2
We give a presentation by generators and relations of a certain monoid
generating a subgroup of index two in the group Aut(F_2) of automorphisms of
the rank two free group F_2 and show that it can be realized as a monoid in the
group B_4 of braids on four strings. In the second part we use Christoffel
words to construct an explicit basis of F_2 lifting any given basis of the free
abelian group Z^2. We further give an algorithm allowing to decide whether two
elements of F_2 form a basis or not. We also show that, under suitable
conditions, a basis has a unique conjugate consisting of two palindromes.Comment: 25 pages, 4 figure
An invariant of tangle cobordisms
We prove that the construction of our previous paper math.QA/0103190 yields
an invariant of tangle cobordisms.Comment: latex, 18 pages, 9 eps figure
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