2000 character limit reached
Connecting essential triangulations I: via 2-3 and 0-2 moves (2405.03539v1)
Published 6 May 2024 in math.GT
Abstract: Suppose that $M$ is a compact, connected three-manifold with boundary. We show that if the universal cover has infinitely many boundary components then $M$ has an ideal triangulation which is essential: no edge can be homotoped into the boundary. Under the same hypotheses, we show that the set of essential triangulations of $M$ is connected via 2-3, 3-2, 0-2, and 2-0 moves. The above results are special cases of our general theory. We introduce $L$-essential triangulations: boundary components of the universal cover receive labels and no edge has the same label at both ends. As an application, under mild conditions on a representation, we construct an ideal triangulation for which a solution to Thurston's gluing equations recovers the given representation. Our results also imply that such triangulations are connected via 2-3, 3-2, 0-2, and 2-0 moves. Together with results of Pandey and Wong, this proves that Dimofte and Garoufalidis' 1-loop invariant is independent of the choice of essential triangulation.
- Gennaro Amendola. A calculus for ideal triangulations of three-manifolds with embedded arcs. Math. Nachr., 278(9):975--994, 2005. doi:10.1002/mana.200310285.
- Invariants of piecewise-linear 3333-manifolds. Trans. Amer. Math. Soc., 348(10):3997--4022, 1996. doi:10.1090/S0002-9947-96-01660-1.
- Alan F. Beardon. The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original.
- Mahler’s measure and the dilogarithm (ii), 2005. arXiv:math/0308041.
- S. Boyer and X. Zhang. On Culler-Shalen seminorms and Dehn filling. Ann. of Math. (2), 148(3):737--801, 1998. doi:10.2307/121031.
- Finding large counterexamples by selectively exploring the Pachner graph. In Erin W. Chambers and Joachim Gudmundsson, editors, 39th International Symposium on Computational Geometry (SoCG 2023), volume 258 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1--21:16, Dagstuhl, Germany, 2023. Schloss Dagstuhl -- Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.SoCG.2023.21.
- Maria Rita Casali. A note about bistellar operations on PL-manifolds with boundary. Geom. Dedicata, 56(3):257--262, 1995. doi:10.1007/BF01263566.
- Marc Culler. The A𝐴Aitalic_A--polynomial and the FFT. Available from http://homepages.math.uic.edu/~culler/talks/apolynomials.pdf, 2005.
- SnapPy, a computer program for studying the geometry and topology of three-manifolds. Available from http://snappy.computop.org (2024-01-01).
- The quantum content of the gluing equations. Geom. Topol., 17(3):1253--1315, 2013. doi:10.2140/gt.2013.17.1253.
- Nathan M. Dunfield. Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds. Invent. Math., 136(3):623--657, 1999. doi:10.1007/s002220050321.
- Pierre Fatou. Théorie des fonctions algébriques. Tome II. Fonctions automorphes. Gauthier-Villars et Cie (Paris), 1930. https://gallica.bnf.fr/ark:/12148/bpt6k927069/.
- From veering triangulations to link spaces and back again, 2022. arXiv:1911.00006.
- Triangulation independent Ptolemy varieties. Math. Z., 289(1-2):663--693, 2018. doi:10.1007/s00209-017-1970-4.
- Gerard Goggin. Global Mobile Media. Taylor & Francis Group, London, United Kingdom, 2010. doi:10.4324/9780203842805.
- C. McA. Gordon. 3333-dimensional topology up to 1960. In History of topology, pages 449--489. North-Holland, Amsterdam, 1999. doi:10.1016/B978-044482375-5/50016-X.
- J. Hempel. 3-Manifolds. AMS Chelsea Publishing Series. AMS Chelsea Pub., American Mathematical Society, 2004. https://books.google.com/books?id=eqcJBAAAQBAJ.
- Triangulations of 3-manifolds with essential edges. Ann. Fac. Sci. Toulouse Math. (6), 24(5):1103--1145, 2015. doi:10.5802/afst.1477.
- A-polynomials, Ptolemy equations and Dehn filling, 2021. arXiv:2002.10356.
- Connecting essential triangulations II: via 2-3 moves only, 2024. In preparation.
- Mikhail Khovanov. sl(3) link homology. Algebr. Geom. Topol., 4:1045--1081, 2004. doi:10.2140/agt.2004.4.1045.
- Felix Klein. Neue Beiträge zur Riemann’schen Functionentheorie. Math. Ann., 21(2):141--218, 1883. doi:10.1007/BF01442920.
- W. B. R. Lickorish. Simplicial moves on complexes and manifolds. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 299--320. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.299.
- Knotinfo: table of knot invariants, 2024. knotinfo.math.indiana.edu/descriptions/a_polynomial.html.
- Thurston’s spinning construction and solutions to the hyperbolic gluing equations for closed hyperbolic 3-manifolds. Proc. Amer. Math. Soc., 141(1):335--350, 2013. doi:10.1090/S0002-9939-2012-11220-1.
- Groups of elliptic linear fractional transformations. Proc. Amer. Math. Soc., 18:1119--1124, 1967. doi:10.2307/2035812.
- Bernard Maskit. On Klein’s combination theorem. Trans. Amer. Math. Soc., 120:499--509, 1965. doi:10.2307/1994540.
- S. Matveev. Algorithmic Topology and Classification of 3-Manifolds. Algorithms and Computation in Mathematics. Springer Berlin Heidelberg, 2007. https://books.google.com/books?id=vFLgAyeVSqAC.
- Edwin E. Moise. Geometric topology in dimensions 2222 and 3333. Graduate Texts in Mathematics, Vol. 47. Springer-Verlag, New York-Heidelberg, 1977.
- Bloch invariants of hyperbolic 3333-manifolds. Duke Math. J., 96(1):29--59, 1999. doi:10.1215/S0012-7094-99-09602-3.
- Udo Pachner. Bistellare Äquivalenz kombinatorischer Mannigfaltigkeiten. Arch. Math. (Basel), 30(1):89--98, 1978. doi:10.1007/BF01226024.
- Geometry of fundamental shadow link complements and applications to the 1-loop conjecture, 2024. arXiv:2308.06643.
- Riccardo Piergallini. Standard moves for standard polyhedra and spines. In III Convegno Nazionale di Topologia, number 18, pages 391--414. Circ. Mat. Palermo, 1988. Third National Conference on Topology (Italian) (Trieste, 1986).
- Jessica S. Purcell. Hyperbolic knot theory, volume 209 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2020. doi:10.1090/gsm/209.
- Traversing three-manifold triangulations and spines. Enseign. Math., 65(1-2):155--206, 2019. doi:10.4171/lem/65-1/2-5.
- Henry Segerman. A generalisation of the deformation variety. Algebr. Geom. Topol., 12(4):2179--2244, 2012. doi:10.2140/agt.2012.12.2179.
- Henry Segerman. Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves. Proc. Amer. Math. Soc., 145(12):5391--5404, 2017. doi:10.1090/proc/13485.
- Pseudo-developing maps for ideal triangulations I: essential edges and generalised hyperbolic gluing equations. In Topology and geometry in dimension three, volume 560 of Contemp. Math., pages 85--102. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/560/11093.
- John M. Sullivan. The geometry of bubbles and foams. In Foams and emulsions (Cargèse, 1997), volume 354 of NATO Adv. Sci. Inst. Ser. E: Appl. Sci., pages 379--402. Kluwer Acad. Publ., Dordrecht, 1999.
- William Thurston. Geometry and topology of 3-manifolds. Available from https://library.slmath.org/nonmsri/gt3m/, 1980.
- William P. Thurston. Hyperbolic structures on 3333-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. (2), 124(2):203--246, 1986. doi:10.2307/1971277.
- State sum invariants of 3333-manifolds and quantum 6j6𝑗6j6 italic_j-symbols. Topology, 31(4):865--902, 1992. doi:10.1016/0040-9383(92)90015-A.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.