648 research outputs found
On the distance between the expressions of a permutation
We prove that the combinatorial distance between any two reduced expressions
of a given permutation of {1, ..., n} in terms of transpositions lies in
O(n^4), a sharp bound. Using a connection with the intersection numbers of
certain curves in van Kampen diagrams, we prove that this bound is sharp, and
give a practical criterion for proving that the derivations provided by the
reversing algorithm of [Dehornoy, JPAA 116 (1997) 115-197] are optimal. We also
show the existence of length l expressions whose reversing requires C l^4
elementary steps
Partial-indistinguishability obfuscation using braids
An obfuscator is an algorithm that translates circuits into
functionally-equivalent similarly-sized circuits that are hard to understand.
Efficient obfuscators would have many applications in cryptography. Until
recently, theoretical progress has mainly been limited to no-go results. Recent
works have proposed the first efficient obfuscation algorithms for classical
logic circuits, based on a notion of indistinguishability against
polynomial-time adversaries. In this work, we propose a new notion of
obfuscation, which we call partial-indistinguishability. This notion is based
on computationally universal groups with efficiently computable normal forms,
and appears to be incomparable with existing definitions. We describe universal
gate sets for both classical and quantum computation, in which our definition
of obfuscation can be met by polynomial-time algorithms. We also discuss some
potential applications to testing quantum computers. We stress that the
cryptographic security of these obfuscators, especially when composed with
translation from other gate sets, remains an open question.Comment: 21 pages,Proceedings of TQC 201
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
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
On definite strongly quasipositive links and L-space branched covers
We investigate the problem of characterising the family of strongly
quasipositive links which have definite symmetrised Seifert forms and apply our
results to the problem of determining when such a link can have an L-space
cyclic branched cover. In particular, we show that if is the dual Garside element and is a strongly quasipositive braid whose braid closure is
definite, then implies that is one of the torus links
or pretzel links . Applying
Theorem 1.1 of our previous paper we deduce that if one of the standard cyclic
branched covers of is an L-space, then is one of
these links. We show by example that there are strongly quasipositive braids
whose closures are definite but not one of these torus or pretzel
links. We also determine the family of definite strongly quasipositive
-braids and show that their closures coincide with the family of strongly
quasipositive -braids with an L-space branched cover.Comment: 62 pages, minor revisions, accepted for publication in Adv. Mat
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
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