SnapPea (http://www.math.uic.edu/~t3m/SnapPy/) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the reader is intimately familiar with all the relevant mathematical background. Does there exist any other document which explains some of the theory in parallel with explaining the code? Failing that, is there a recommended source which just explains the theory, but in a way which meshes nicely with the code?
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asked Sep 22, 2015 at 14:48
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2$\begingroup$ A small comment that you might be aware of. The code for SnapPy has fairly extensive documentation in it, explaining many of the key theoretical steps, such as how the software finds the hyperbolic structures. It pretty much only assumes familiarity with thinking of the Riemann sphere as the boundary of hyperbolic 3-space. $\endgroup$– Ryan BudneyCommented Sep 22, 2015 at 19:16
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2$\begingroup$ One further comment, the figure-8 knot exterior example in Thurston's lecture notes is really the key example that shows you how SnapPea works. I have another exposition of it (that some people prefer) that I could send you if you like. $\endgroup$– Ryan BudneyCommented Sep 22, 2015 at 19:19
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$\begingroup$ @RyanBudney: it would be great if you could send me that, thanks. $\endgroup$– Neil StricklandCommented Sep 22, 2015 at 20:29
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4$\begingroup$ In the March 2015 version of my notes it starts on page 131. rybu.org/DGNotes Any feedback is welcome. These notes certainly could use some polishing... $\endgroup$– Ryan BudneyCommented Sep 22, 2015 at 20:37
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$\begingroup$ I found that reading Jeff Weeks PhD thesis was very helpful. Also: lovely hand-drawn diagrams! $\endgroup$– Sam NeadCommented Apr 23, 2017 at 19:00
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2 Answers
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First as pointed out in the comments, the documentation of the SnapPea kernel (now maintained as SnapPy) is extensive. It contains theorems and proofs as well as a fairly thorough treatment many of the functions of SnapPea/SnapPy.
Also, in addition to Thurston's notes, one might also consider Section E.6 of Benedetti and Petronio's Lectures on Hyperbolic Geometry as it provides a good explanation of the polynomial equations that one must solve to find a hyperbolic structure associated to the triangulation.
However, the question asks for theory that is most closely connected to the code, on that note, there is also Weeks' article in the Handbook of Knot Theory:
Weeks, Jeff. "Computation of hyperbolic structures in knot theory." Handbook of Knot Theory (2005): 461-480. http://arxiv.org/abs/math/0309407
This article mainly focuses on the algorithm to triangulate the complement of a knot or link and then solve a relevant system of equations (coming from Thurston), before finishing with a discussion of how the program considers Dehn filling. As Carlo Beenakker points out, other aspects of the program, such as the computation isometry group (via a computation of the canonical triangulation) are covered elsewhere in Jeff Weeks' works.
answered Sep 24, 2015 at 0:43
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Since SnapPea was developed to accompany William Thurston's lecture notes on hyperbolic three-manifolds, these might make a good background reading. Applications of SnapPea’s isometry checking algorithm are described by Jeffrey Weeks in this 1993 article.
Hyperbolic 3-manifolds have proven to be a rich and interesting field of mathematics. Because hyperbolic structures may be computed by hand only in the very simplest examples, computer calculations are essential in any systematic study. The computer program SnapPea creates hyperbolic 3-manifolds and computes various graphical, algebraic and numerical invariants. In this article we present a simple and surprising theorem which underlies SnapPea’s algorithm for determining whether two cusped hyperbolic 3-manifolds are isometric. We review the relationship between closed and cusped hyperbolic 3-manifolds, describes SnapPea’s approach to checking for isometries, and give several applications.
answered Sep 22, 2015 at 17:17
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5$\begingroup$ I downloaded the lecture notes, but they come to 373 pages in total. Call me a wimp, but I was hoping for something a little lighter. $\endgroup$ Commented Sep 22, 2015 at 19:12
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$\begingroup$ well, there's the "notes on the notes" and even an "intro to the notes on the notes", to makes this more digestible: ebooks.cambridge.org/… --- math.lsa.umich.edu/~canary/foreword.pdf $\endgroup$ Commented Sep 22, 2015 at 20:35
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2$\begingroup$ @CarloBeenakker those things tend to focus on other aspects of the Thurston note than those relevant to SnapPea $\endgroup$ Commented Sep 22, 2015 at 21:08