410,455 research outputs found
The complexity of the normal surface solution space
Normal surface theory is a central tool in algorithmic three-dimensional
topology, and the enumeration of vertex normal surfaces is the computational
bottleneck in many important algorithms. However, it is not well understood how
the number of such surfaces grows in relation to the size of the underlying
triangulation. Here we address this problem in both theory and practice. In
theory, we tighten the exponential upper bound substantially; furthermore, we
construct pathological triangulations that prove an exponential bound to be
unavoidable. In practice, we undertake a comprehensive analysis of millions of
triangulations and find that in general the number of vertex normal surfaces is
remarkably small, with strong evidence that our pathological triangulations may
in fact be the worst case scenarios. This analysis is the first of its kind,
and the striking behaviour that we observe has important implications for the
feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2
tables; v2: added minor clarification
Maximal admissible faces and asymptotic bounds for the normal surface solution space
The enumeration of normal surfaces is a key bottleneck in computational
three-dimensional topology. The underlying procedure is the enumeration of
admissible vertices of a high-dimensional polytope, where admissibility is a
powerful but non-linear and non-convex constraint. The main results of this
paper are significant improvements upon the best known asymptotic bounds on the
number of admissible vertices, using polytopes in both the standard normal
surface coordinate system and the streamlined quadrilateral coordinate system.
To achieve these results we examine the layout of admissible points within
these polytopes. We show that these points correspond to well-behaved
substructures of the face lattice, and we study properties of the corresponding
"admissible faces". Key lemmata include upper bounds on the number of maximal
admissible faces of each dimension, and a bijection between the maximal
admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in
Journal of Combinatorial Theory A
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
Converting between quadrilateral and standard solution sets in normal surface theory
The enumeration of normal surfaces is a crucial but very slow operation in
algorithmic 3-manifold topology. At the heart of this operation is a polytope
vertex enumeration in a high-dimensional space (standard coordinates).
Tollefson's Q-theory speeds up this operation by using a much smaller space
(quadrilateral coordinates), at the cost of a reduced solution set that might
not always be sufficient for our needs. In this paper we present algorithms for
converting between solution sets in quadrilateral and standard coordinates. As
a consequence we obtain a new algorithm for enumerating all standard vertex
normal surfaces, yielding both the speed of quadrilateral coordinates and the
wider applicability of standard coordinates. Experimentation with the software
package Regina shows this new algorithm to be extremely fast in practice,
improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the
journal styl
Involutions of polynomially parametrized surfaces
We provide an algorithm for detecting the involutions leaving a surface
defined by a polynomial parametrization invariant. As a consequence, the
symmetry axes, symmetry planes and symmetry center of the surface, if any, can
be determined directly from the parametrization, without computing or making
use of the implicit representation. The algorithm is based on the fact, proven
in the paper, that any involution of the surface comes from an involution of
the parameter space (the real plane, in our case); therefore, by determining
the latter, the former can be found. The algorithm has been implemented in the
computer algebra system Maple 17. Evidence of its efficiency for moderate
degrees, examples and a complexity analysis are also given
WdW-patches in AdS and complexity change under conformal transformations II
We study the null-boundaries of Wheeler-de Witt (WdW) patches in three
dimensional Poincare-AdS, when the selected boundary timeslice is an arbitrary
(non-constant) function, presenting some useful analytic statements about them.
Special attention will be given to the piecewise smooth nature of the
null-boundaries, due to the emergence of caustics and null-null joint curves.
This is then applied, in the spirit of our previous paper arXiv:1806.08376, to
the problem of how complexity of the CFT groundstate changes under a small
local conformal transformation according to the action (CA) proposal. In stark
contrast to the volume (CV) proposal, where this change is only proportional to
the second order in the infinitesimal expansion parameter , we show
that in the CA case we obtain terms of order and even
. This has strong implications for the possible
field-theory duals of the CA proposal, ruling out an entire class of them.Comment: 31 pages + appendices, 9 figures v2: minor improvements, matches
published versio
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