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Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves


Author: Henry Segerman
Journal: Proc. Amer. Math. Soc. 145 (2017), 5391-5404
MSC (2010): Primary 57Q15; Secondary 57M27
DOI: https://doi.org/10.1090/proc/13485
Published electronically: August 29, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the Pachner graph of such triangulations is connected. In this paper, we extend this result to show that (again with a small number of known exceptions) the subgraph of the Pachner graph consisting of triangulations without degree one edges is also connected for single-vertex triangulations of closed manifolds and ideal triangulations of manifolds with non-spherical boundary components.


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  • [1] Blake Dadd and Aochen Duan, Constructing infinitely many geometric triangulations of the figure eight knot complement, Proc. Amer. Math. Soc. 144 (2016), no. 10, 4545-4555. MR 3531201, https://doi.org/10.1090/proc/13076
  • [2] Tudor Dimofte, Davide Gaiotto, and Sergei Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013), no. 5, 975-1076. MR 3262519
  • [3] Tudor Dimofte, Davide Gaiotto, and Sergei Gukov, Gauge theories labelled by three-manifolds, Comm. Math. Phys. 325 (2014), no. 2, 367-419. MR 3148093, https://doi.org/10.1007/s00220-013-1863-2
  • [4] Tudor Dimofte and Stavros Garoufalidis, The quantum content of the gluing equations, Geom. Topol. 17 (2013), no. 3, 1253-1315. MR 3073925, https://doi.org/10.2140/gt.2013.17.1253
  • [5] Stavros Garoufalidis, Craig D. Hodgson, J. Hyam Rubinstein, and Henry Segerman, 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, Geom. Topol. 19 (2015), no. 5, 2619-2689. MR 3416111, https://doi.org/10.2140/gt.2015.19.2619
  • [6] S. V. Matveev, Transformations of special spines, and the Zeeman conjecture, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 5, 1104-1116, 1119 (Russian); English transl., Math. USSR-Izv. 31 (1988), no. 2, 423-434. MR 925096
  • [7] Sergei Matveev, Algorithmic topology and classification of 3-manifolds, 2nd ed., Algorithms and Computation in Mathematics, vol. 9, Springer, Berlin, 2007. MR 2341532
  • [8] Riccardo Piergallini, Standard moves for standard polyhedra and spines, Third National Conference on Topology (Italian) (Trieste, 1986), Rend. Circ. Mat. Palermo (2) Suppl. 18 (1988), 391-414. Third National Conference on Topology (Italian) (Trieste, 1986). MR 958750
  • [9] Stavros Garoufalidis, The 3D index of an ideal triangulation and angle structures, Ramanujan J. 40 (2016), no. 3, 573-604. With an appendix by Sander Zwegers. MR 3522084, https://doi.org/10.1007/s11139-016-9771-7

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Additional Information

Henry Segerman
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: segerman@math.okstate.edu

DOI: https://doi.org/10.1090/proc/13485
Received by editor(s): June 3, 2016
Received by editor(s) in revised form: September 18, 2016
Published electronically: August 29, 2017
Additional Notes: The author was supported in part by National Science Foundation grant DMS-1308767.
Communicated by: David Futer
Article copyright: © Copyright 2017 American Mathematical Society

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