Verified Computations for Hyperbolic 3-Manifolds

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method is an implementation of interval arithmetic and Krawczyk’s test. These techniques represent an improvement over existing algorithms as they are faster while accounting for error accumulation in a more direct and user-friendly way.

[1]  J. Morgan,et al.  Recent progress on the Poincaré conjecture and the classification of 3-manifolds , 2004 .

[2]  Bounds on exceptional Dehn filling , 1999, math/9906183.

[3]  T. Sunaga.  Theory of an interval algebra and its application to numerical analysis , 2009 .

[4]  Siegfried M. Rump,et al.  Verification methods: rigorous results using floating-point arithmetic , 2010, Acta Numerica.

[5]  D. Epstein,et al.  Three-dimensional manifolds , 1960 .

[6]  Hidetoshi Masai,et al.  Exceptional surgeries on alternating knots , 2013, 1310.3472.

[7]  Richard Hughes,et al.  Geometry and Topology of 3-manifolds , 2011 .

[8]  Abe Shenitzer,et al.  The Poincaré Conjecture? , 2006, Am. Math. Mon..

[9]  Benjamin A. Burton.  The cusped hyperbolic census is complete , 2014, ArXiv.

[10]  J. Milnor.  Hyperbolic geometry: The first 150 years , 1982 .

[11]  Martin Hildebrand,et al.  A Computer Generated Census of Cusped Hyperbolic 3-Manifolds , 1989, Computers and Mathematics.

[12]  C. Petronio,et al.  Exceptional Dehn surgery on the minimally twisted five-chain link , 2011, 1109.0903.

[13]  Word hyperbolic Dehn surgery , 1998, math/9808120.

[14]  Harriet H. Moser.  Proving a manifold to be hyperbolic once it has been approximated to be so , 2008, 0809.1203.

[15]  Ying‐Qing Wu.  Persistently laminar branched surfaces , 2010, 1008.2680.

[16]  Jeffrey R. Weeks,et al.  A census of cusped hyperbolic 3-manifolds , 1999, Math. Comput..

[17]  Jeffrey R. Weeks,et al.  Symmetries, Isometries and Length Spectra of Closed Hyperbolic Three-Manifolds , 1994, Exp. Math..

[19]  J. Morgan,et al.  Ricci Flow and the Poincare Conjecture , 2006, math/0607607.

[20]  Louis B. Rall,et al.  Automatic Differentiation: Techniques and Applications , 1981, Lecture Notes in Computer Science.

[21]  Richard D. Neidinger,et al.  Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming , 2010, SIAM Rev..

[22]  Kohshi Okumura,et al.  The Contribution of T. Sunaga to Interval Analysis and Reliable Computing , 1998, SCAN.

[23]  Rudolf Krawczyk,et al.  Fehlerabschätzung reeller Eigenwerte und Eigenvektoren von Matrizen , 1969, Computing.

[24]  Small curvature surfaces in hyperbolic 3-manifolds , 2004, math/0409455.

[25]  W. Thurston.  Three dimensional manifolds, Kleinian groups and hyperbolic geometry , 1982 .

[26]  G. Robert Meyerhoff,et al.  Homotopy hyperbolic 3-manifolds are hyperbolic , 1996 .

[27]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[28]  Rudolf Krawczyk,et al.  Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken , 1969, Computing.

[29]  R. Benedetti,et al.  Lectures on Hyperbolic Geometry , 1992 .

[30]  Walter D. Neumann,et al.  Computing Arithmetic Invariants of 3-Manifolds , 2000, Exp. Math..

[31]  Siddhartha Gadgil,et al.  Ricci Flow and the Poincaré Conjecture , 2007 .

[32]  G. Perelman.  Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[33]  Don Zagier,et al.  Volumes of hyperbolic three-manifolds , 1985 .