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Computation of hyperbolic structures in knot theory. (English) Zbl 1096.57015

Menasco, William (ed.) et al., Handbook of knot theory. Amsterdam: Elsevier (ISBN 0-444-51452-X/hbk). 461-480 (2005).
Geometrical structures (typically hyperbolic) provide deep insight into the topology of knot and link complements. This survey shows how to construct a hyperbolic structure on the complement of a topologically defined knot or link and how to perform hyperbolic Dehn filling.
The article begins with the geometry of 2-dimensional link complements to provide an overview of all the main ideas. Then the author explains an efficient algorithm for triangulating 3-dimensional link complements, shows how to compute the hyperbolic structure and finally shows how the hyperbolic structure deforms to yield hyperbolic structures on closed manifolds obtained by Dehn filling.
The computer program SnapPea implements the applications based on the foundation described in the present paper. The SnapPea source code contains detailed explanations of all algorithms used (algorithms for triangulating a link complement efficiently and for converging quickly to the hyperbolic structure while avoiding singularities in the parameter space).
For the entire collection see [Zbl 1073.57001].

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)

Software:

SnapPea
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