42 research outputs found
Identifying lens spaces in polynomial time
We show that if a closed, oriented 3-manifold M is promised to be
homeomorphic to a lens space L(n,k) with n and k unknown, then we can compute
both n and k in polynomial time in the size of the triangulation of M. The
tricky part is the parameter k. The idea of the algorithm is to calculate
Reidemeister torsion using numerical analysis over the complex numbers, rather
than working directly in a cyclotomic field.Comment: 5 pages. A major revision with a new title, and with a classical
algorithm rather than a quantum algorith
A combinatorial approach to knot recognition
This is a report on our ongoing research on a combinatorial approach to knot
recognition, using coloring of knots by certain algebraic objects called
quandles. The aim of the paper is to summarize the mathematical theory of knot
coloring in a compact, accessible manner, and to show how to use it for
computational purposes. In particular, we address how to determine colorability
of a knot, and propose to use SAT solving to search for colorings. The
computational complexity of the problem, both in theory and in our
implementation, is discussed. In the last part, we explain how coloring can be
utilized in knot recognition
Rectangular knot diagrams classification with deep learning
In this article we discuss applications of neural networks to recognising
knots and, in particular, to the unknotting problem. One of motivations for
this study is to understand how neural networks work on the example of a
problem for which rigorous mathematical algorithms for its solution are known.
We represent knots by rectangular Dynnikov diagrams and apply neural networks
to distinguish a given diagram class from the given finite families of
topological types. The data presented to the program is generated by applying
Dynnikov moves to initial samples. The significance of using these diagrams and
moves is that in this context the problem of determining whether a diagram is
unknotted is a finite search of a bounded combinatorial space.Comment: 23 pages, 16 figures, LaTeX documen
2-manifold recognition is in logspace
We prove that the homeomorphism problem for 2 manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected s, t-connectivity problem in graphs
2-manifold recognition is in logspace
We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected -connectivity problem in graphs
The complexity of detecting taut angle structures on triangulations
There are many fundamental algorithmic problems on triangulated 3-manifolds
whose complexities are unknown. Here we study the problem of finding a taut
angle structure on a 3-manifold triangulation, whose existence has implications
for both the geometry and combinatorics of the triangulation. We prove that
detecting taut angle structures is NP-complete, but also fixed-parameter
tractable in the treewidth of the face pairing graph of the triangulation.
These results have deeper implications: the core techniques can serve as a
launching point for approaching decision problems such as unknot recognition
and prime decomposition of 3-manifolds.Comment: 22 pages, 10 figures, 3 tables; v2: minor updates. To appear in SODA
2013: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete
Algorithm