420 research outputs found
Combinatorics of normal sequences of braids
Many natural counting problems arise in connection with the normal form of
braids--and seem to have never been considered so far. Here we solve some of
them by analysing the normality condition in terms of the associated
permutations, their descents and the corresponding partitions. A number of
different induction schemes appear in that framework
A divisibility result on combinatorics of generalized braids
For every finite Coxeter group , each positive braids in the
corresponding braid group admits a unique decomposition as a finite sequence of
elements of , the so-called Garside-normal form.The study of the
associated adjacency matrix allows to count the number of
Garside-normal form of a given length.In this paper we prove that the
characteristic polynomial of divides the one of . The
key point is the use of a Hopf algebra based on signed permutations. A similar
result was already known for the type . We observe that this does not hold
for type . The other Coxeter types (, , and ) are also studied.Comment: 28 page
Still another approach to the braid ordering
We develop a new approach to the linear ordering of the braid group ,
based on investigating its restriction to the set \Div(\Delta\_n^d) of all
divisors of in the monoid , i.e., to positive
-braids whose normal form has length at most . In the general case, we
compute several numerical parameters attached with the finite orders
(\Div(\Delta\_n^d), <). In the case of 3 strands, we moreover give a complete
description of the increasing enumeration of (\Div(\Delta\_3^d), <). We
deduce a new and specially direct construction of the ordering on , and a
new proof of the result that its restriction to is a well-ordering of
ordinal type
Unprovability results involving braids
We construct long sequences of braids that are descending with respect to the
standard order of braids (``Dehornoy order''), and we deduce that, contrary to
all usual algebraic properties of braids, certain simple combinatorial
statements involving the braid order are true, but not provable in the
subsystems ISigma1 or ISigma2 of the standard Peano system.Comment: 32 page
Laver's results and low-dimensional topology
In connection with his interest in selfdistributive algebra, Richard Laver
established two deep results with potential applications in low-dimensional
topology, namely the existence of what is now known as the Laver tables and the
well-foundedness of the standard ordering of positive braids. Here we present
these results and discuss the way they could be used in topological
applications
Operads within monoidal pseudo algebras
A general notion of operad is given, which includes as instances, the operads
originally conceived to study loop spaces, as well as the higher operads that
arise in the globular approach to higher dimensional algebra. In the framework
of this paper, one can also describe symmetric and braided analogues of higher
operads, likely to be important to the study of weakly symmetric, higher
dimensional monoidal structures
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