SnapPea
Encyclopedia
Free software
Free software, software libre or libre software is software that can be used, studied, and modified without restriction, and which can be copied and redistributed in modified or unmodified form either without restriction, or with restrictions that only ensure that further recipients can also do...
designed to help mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s, in particular low-dimensional topologists
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
, study hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
s. The primary developer is Jeffrey Weeks
Jeffrey Weeks (mathematician)
Jeffrey Renwick Weeks is an American mathematician, a geometric topologist and cosmologist.-Biography:Weeks received his B.A. from Dartmouth College in 1978, and his Ph.D. in mathematics from Princeton University in 1985, under the supervision of William Thurston...
, who created the first version as part of his doctoral thesis, supervised by William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
. The latest version is 3.0d3. Marc Culler
Marc Culler
Marc Edward Culler is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the...
and Nathan Dunfield have a version which extends SnapPea's Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
interface (see external links below).
The following people are credited in SnapPea 2.5.3's list of acknowledgments: Colin Adams
Colin Adams (mathematician)
Colin Conrad Adams is a mathematician primarily working in the areas of hyperbolic 3-manifolds and knot theory. His book, The Knot Book, has been praised for its accessible approach to advanced topics in knot theory. He is currently Francis Christopher Oakley Third Century Professor of...
, Bill Arveson, Pat Callahan, Joe Christy, Dave Gabai
David Gabai
David Gabai, a mathematician, is currently a professor at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects....
, Charlie Gunn, Martin Hildebrand, Craig Hodgson, Diane Hoffoss, A. C. Manoharan, Al Marden, Dick McGehee, Rob Meyerhoff, Lee Mosher, Walter Neumann, Carlo Petronio, Mark Phillips, Alan Reid, and Makoto Sakuma.
The C
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
source code is extensively commented by Weeks and contains useful descriptions of the mathematics involved with references.
Algorithms and functions
At the core of SnapPea are two main algorithms. The first attempts to find a minimal ideal triangulation of a given link complement. The second computes the canonical decomposition of a cusped hyperbolic 3-manifoldHyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
. Almost all the other functions of SnapPea rely in some way on one of these decompositions.
Minimal ideal triangulation
SnapPea inputs data in a variety of formats. Given a link diagram, SnapPea can ideally triangulate the link complement. It then performs a sequence of simplifications to find a minimal ideal triangulation.Once a minimal ideal triangulation is found, SnapPea can try and find a hyperbolic structure. In his Princeton lecture notes, Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
noted a method for describing the geometric shape of each hyperbolic tetrahedron by a complex number and a set of nonlinear equations of complex variables whose solution would give a complete hyperbolic metric on the 3-manifold. These equations consist of edge equations and cusp (completeness) equations. SnapPea uses an iterative method utilizing Newton's method
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
to search for solutions. If no solution exists, then it retriangulated randomly, repeating the process.
The minimality of the triangulation is meant to increase the likelihood that such a solution exists, since heuristically one might expect the minimal triangulation to be "straightened" without causing degenerations or overlapping of tetrahedra.
From this description of the hyperbolic structure on a link complement, SnapPea can then perform hyperbolic Dehn filling
Hyperbolic Dehn surgery
In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold...
on the cusps to obtain more hyperbolic 3-manifolds. SnapPea does this by taking any given slopes which determine certain Dehn filling equations (also explained in Thurston's notes), and then adjusting the shapes of the ideal tetrahedra to give solutions to these equations and the edge equations. This gives an (incomplete) hyperbolic structure on almost all of the Dehn-filled manifold. The completion gives a hyperbolic structure on the entire manifold. Its volume is the sum of the volumes of the adjusted tetrahedra.
Canonical decomposition
SnapPea is usually able to compute the canonical decomposition of a cusped hyperbolic 3-manifold from a given ideal triangulation. If not, then it randomly retriangulates and tries again. This has never been known to fail.The canonical decomposition allows SnapPea to tell two cusped hyperbolic 3-manifolds apart by turning the problem of recognition into a combinatorial question, i.e. checking if the two manifolds have combinatorially equivalent canonical decompositions. SnapPea is also able to check if two closed hyperbolic 3-manifolds are isometric by drilling out short geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s to create cusped hyperbolic 3-manifolds and then using the canonical decomposition as before.
The recognition algorithm allow SnapPea to tell two hyperbolic knots or links apart. Weeks, et al., were also able to compile different censuses of hyperbolic 3-manifolds by using the algorithm to cull lists of duplicates.
Additionally, from the canonical decomposition, SnapPea is able to:
- Compute the Ford domain
- Compute the symmetry group
Censuses
SnapPea has several databases of hyperbolic 3-manifolds available for systematic study.- Cusped census
- Closed census
See also
- ReginaRegina (program)Regina is a suite of mathematical software for 3-manifold topologists. It focuses upon the study of 3-manifold triangulations and includes support for normal surfaces and angle structures.- Features :...
incorporates aspects of SnapPea. - Damian Heard's Orb allows hyperbolic manifolds with totally geodesic boundary, and also orbifolds where the orbifold locus contains trivalent vertices.
- Computational topology