Practical bounds for a Dehn parental test
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- by Robert C. Haraway III PDF
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Abstract:
Hodgson and Kerckhoff proved a powerful theorem, half of which they used to make Thurston’s Dehn surgery theorem effective. The calculations derived here use both halves of Hodgson and Kerckhoff’s theorem to give bounds leading towards a practical algorithm to tell, given two orientable complete hyperbolic 3-manifolds $M,N$ of finite volume, whether or not $N$ is a Dehn filling of $M.$References
- Ian Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000), 431–449. MR 1799796, DOI 10.2140/gt.2000.4.431
- Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1053–1077. With an appendix by Nathan Dunfield. MR 2328715, DOI 10.1090/S0894-0347-07-00564-4
- Christopher K. Atkinson and David Futer, The lowest volume 3-orbifolds with high torsion, Trans. Amer. Math. Soc. 369 (2017), no. 8, 5809–5827. MR 3646779, DOI 10.1090/tran/6920
- David Coulson, Oliver A. Goodman, Craig D. Hodgson, and Walter D. Neumann, Computing arithmetic invariants of 3-manifolds, Experiment. Math. 9 (2000), no. 1, 127–152. MR 1758805
- Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks, SnapPy, a computer program for studying the geometry and topology of $3$-manifolds, available at https://snappy.computop.org/ (22/02/2018).
- The Coq development team, The Coq reference manual, v.8.7.2, 2017, available at https://coq.inria.fr/distrib/current/refman/ (22/02/2018).
- David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Dehn filling, volume, and the Jones polynomial, J. Differential Geom. 78 (2008), no. 3, 429–464. MR 2396249
- David Gabai, Robert Meyerhoff, and Peter Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009), no. 4, 1157–1215. MR 2525782, DOI 10.1090/S0894-0347-09-00639-0
- David Gabai, Robert Meyerhoff, and Peter Milley, Mom technology and volumes of hyperbolic 3-manifolds, Comment. Math. Helv. 86 (2011), no. 1, 145–188. MR 2745279, DOI 10.4171/CMH/221
- Craig Hodgson and Hidetoshi Masai, On the number of hyperbolic 3-manifolds of a given volume, Geometry and topology down under, Contemp. Math., vol. 597, Amer. Math. Soc., Providence, RI, 2013, pp. 295–320. MR 3186679, DOI 10.1090/conm/597/11767
- Craig D. Hodgson and Steven P. Kerckhoff, The shape of hyperbolic Dehn surgery space, Geom. Topol. 12 (2008), no. 2, 1033–1090. MR 2403805, DOI 10.2140/gt.2008.12.1033
- Craig D. Hodgson and Jeffrey R. Weeks, Symmetries, isometries and length spectra of closed hyperbolic three-manifolds, Experiment. Math. 3 (1994), no. 4, 261–274. MR 1341719
- Neil Hoffman, Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin’ichi Oishi, and Akitoshi Takayasu, Verified computations for hyperbolic 3-manifolds, Exp. Math. 25 (2016), no. 1, 66–78. MR 3424833, DOI 10.1080/10586458.2015.1029599
- Neil R. Hoffman and Genevieve S. Walsh, The big Dehn surgery graph and the link of $S^3$, Proc. Amer. Math. Soc. Ser. B 2 (2015), 17–34. MR 3422666, DOI 10.1090/bproc/20
- Robert C. Haraway III, GitHub repository, available at https://github.com/bobbycyiii/cheeky/ (22/02/2018).
- Greg Kuperberg, Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization, https://arxiv.org/abs/1508.06720 [math.GT], 2015.
- Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243–282. MR 1756996, DOI 10.1007/s002220000047
- Jason Manning, Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem, Geom. Topol. 6 (2002), 1–25. MR 1885587, DOI 10.2140/gt.2002.6.1
- Guillaume Melquiond, Interval package for Coq, available at http://coq-interval.gforge.inria.fr (22/02/2018).
- Peter Milley, Minimum volume hyperbolic 3-manifolds, J. Topol. 2 (2009), no. 1, 181–192. MR 2499442, DOI 10.1112/jtopol/jtp006
- Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI 10.1016/0040-9383(85)90004-7
- Peter Scott and Hamish Short, The homeomorphism problem for closed 3-manifolds, Algebr. Geom. Topol. 14 (2014), no. 4, 2431–2444. MR 3331689, DOI 10.2140/agt.2014.14.2431
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
Additional Information
- Robert C. Haraway III
- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
- MR Author ID: 1297918
- ORCID: 0000-0002-0952-1393
- Email: robert.haraway@okstate.edu
- Received by editor(s): November 3, 2016
- Received by editor(s) in revised form: December 20, 2017, February 22, 2018, March 21, 2018, April 2, 2018, and April 29, 2018
- Published electronically: October 18, 2018
- Additional Notes: This research was supported in part by NSF grant DMS-1006553.
- Communicated by: David Futer
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 427-442
- MSC (2010): Primary 57M50; Secondary 57-04
- DOI: https://doi.org/10.1090/proc/14207
- MathSciNet review: 3876760