6,330 research outputs found
Composition problems for braids: Membership, Identity and Freeness
In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with three strands, , have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-complete for braids with only three strands. The membership problem is decidable in NP for , but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least five strands, but decidability of these problems for remains open. Finally we show that the freeness problem for semigroups of braids from is also decidable in NP. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms
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 Garside-theoretic approach to the reducibility problem in braid groups
Let denote the -punctured disk in the complex plane, where the
punctures are on the real axis. An -braid is said to be
\emph{reducible} if there exists an essential curve system \C in ,
called a \emph{reduction system} of , such that \alpha*\C=\C where
\alpha*\C denotes the action of the braid on the curve system \C.
A curve system \C in is said to be \emph{standard} if each of its
components is isotopic to a round circle centered at the real axis.
In this paper, we study the characteristics of the braids sending a curve
system to a standard curve system, and then the characteristics of the
conjugacy classes of reducible braids. For an essential curve system \C in
, we define the \emph{standardizer} of \C as \St(\C)=\{P\in
B_n^+:P*\C{is standard}\} and show that \St(\C) is a sublattice of .
In particular, there exists a unique minimal element in \St(\C). Exploiting
the minimal elements of standardizers together with canonical reduction systems
of reducible braids, we define the outermost component of reducible braids, and
then show that, for the reducible braids whose outermost component is simpler
than the whole braid (including split braids), each element of its ultra summit
set has a standard reduction system. This implies that, for such braids,
finding a reduction system is as easy as finding a single element of the ultra
summit set.Comment: 38 pages, 18 figures, published versio
Vassiliev-Kontsevich invariants and Parseval's theorem
We use an example to provide evidence for the statement: the
Vassiliev-Kontsevich invariants of a knot (or braid) can be redefined
so that . This constructs a knot from its
Vassiliev-Kontsevich invariants, like a power series expansion. The example is
pure braids on two strands , which leads to solving
for a Laurent series in . We set and use Parseval's theorem for Fourier series to
prove . Finally we describe some problems, particularly a Plancherel
theorem for braid groups, whose solution would take us towards a proof of
.Comment: 5 pages, 2 figures. Extensively revised. Discussion of extending
result to braids on more strands and to knots added. Two figures adde
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