834 research outputs found
The twistor discriminant locus of the Fermat cubic
We consider the discriminant locus of the Fermat cubic under the twistor
fibration . We show that it has a conformal symmetry
group of order and use this to identify its topology.Comment: 30 pages, 4 figure
Real algebraic surfaces with isolated singularities
Given a real algebraic surface S in RP3, we propose a constructive procedure to determine the topology of S and to compute non-trivial topological invariants for the pair (RP3, S) under the hypothesis that the real singularities of S are isolated. In particular, starting from an implicit equation of the surface, we compute the number of connected components of S, their Euler characteristics and the weighted 2-adjacency graph of the surface
Open String Diagrams I: Topological Type
An arbitrary Feynman graph for string field theory interactions is analysed
and the homeomorphism type of the corresponding world sheet surface is
completely determined even in the non-orientable cases. Algorithms are found to
mechanically compute the topological characteristics of the resulting surface
from the structure of the signed oriented graph. Whitney's
permutation-theoretic coding of graphs is utilized
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
The homeomorphism problem for closed 3-manifolds
We give a more geometric approach to an algorithm for deciding whether two
hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach
to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.Comment: first version: 12 pages. Replacement: 14 pages. Includes minor
improvements to exposition in response to referee's comment
Decision problems for 3-manifolds and their fundamental groups
We survey the status of some decision problems for 3-manifolds and their
fundamental groups. This includes the classical decision problems for finitely
presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and
also the Homeomorphism Problem for 3-manifolds and the Membership Problem for
3-manifold groups.Comment: 31 pages, final versio
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