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The complement of the figure-eight knot geometrically bounds


Author: Leone Slavich
Journal: Proc. Amer. Math. Soc. 145 (2017), 1275-1285
MSC (2010): Primary 51M10, 51M15, 51M20, 52B11
DOI: https://doi.org/10.1090/proc/13272
Published electronically: August 30, 2016
MathSciNet review: 3589325
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Abstract: We show that some hyperbolic $ 3$-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron are geodesically embedded in a complete, finite volume, hyperbolic $ 4$-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic $ 4$-manifold. This is the first example of geometrically bounding hyperbolic knot complements and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume.


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  • [1] Colin Adams and William Sherman, Minimum ideal triangulations of hyperbolic $ 3$-manifolds, Discrete Comput. Geom. 6 (1991), no. 2, 135-153. MR 1083629, https://doi.org/10.1007/BF02574680
  • [2] M. Culler, N. M. Dunfield, and J. R. Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds, http://snappy.computop.org
  • [3] E. Fominykh, S. Garoufalidis, M. Goerner, V. Tarkaev, and A. Vesnin, A census of tetrahedral hyperbolic manifolds, arXiv:1502.00383
  • [4] G. W. Gibbons, Tunnelling with a negative cosmological constant, Nuclear Phys. B 472 (1996), no. 3, 683-708. MR 1399279, https://doi.org/10.1016/0550-3213(96)00207-6
  • [5] M. Goerner: A census of hyperbolic Platonic manifolds and augmented knotted trivalent graphs, arXiv:1602.02208
  • [6] M. Gromov and I. Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. 66 (1988), 93-103. MR 932135
  • [7] A. Kolpakov, L. Slavich: Symmetries of hyperbolic $ 4$-manifolds, Int. Math. Res. Notices first published online July 25, 2015, doi:10.1093/imrn/rnv210, arXiv:1409.1910.
  • [8] D. D. Long and A. W. Reid, On the geometric boundaries of hyperbolic $ 4$-manifolds, Geom. Topol. 4 (2000), 171-178 (electronic). MR 1769269, https://doi.org/10.2140/gt.2000.4.171
  • [9] D. D. Long and A. W. Reid, Constructing hyperbolic manifolds which bound geometrically, Math. Res. Lett. 8 (2001), no. 4, 443-455. MR 1849261, https://doi.org/10.4310/MRL.2001.v8.n4.a5
  • [10] B. Martelli, Hyperbolic $ 3$-manifolds that embed geodesically, arXiv:1510.06325.
  • [11] Alexander Kolpakov, Bruno Martelli, and Steven Tschantz, Some hyperbolic three-manifolds that bound geometrically, Proc. Amer. Math. Soc. 143 (2015), no. 9, 4103-4111. MR 3359598, https://doi.org/10.1090/proc/12520
  • [12] John G. Ratcliffe and Steven T. Tschantz, Gravitational instantons of constant curvature, Classical Quantum Gravity 15 (1998), no. 9, 2613-2627. Topology of the Universe Conference (Cleveland, OH, 1997). MR 1649662, https://doi.org/10.1088/0264-9381/15/9/009
  • [13] John G. Ratcliffe and Steven T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000), no. 1, 101-125. MR 1758804
  • [14] John G. Ratcliffe and Steven T. Tschantz, On the growth of the number of hyperbolic gravitational instantons with respect to volume, Classical Quantum Gravity 17 (2000), no. 15, 2999-3007. MR 1777000, https://doi.org/10.1088/0264-9381/17/15/310
  • [15] Alan W. Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991), no. 1, 171-184. MR 1099096, https://doi.org/10.1112/jlms/s2-43.1.171
  • [16] Leone Slavich, A geometrically bounding hyperbolic link complement, Algebr. Geom. Topol. 15 (2015), no. 2, 1175-1197. MR 3342689, https://doi.org/10.2140/agt.2015.15.1175
  • [17] W. P. Thurston, The Geometry and Topology of $ 3$-manifolds, mimeographed notes, Princeton, 1979

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Additional Information

Leone Slavich
Affiliation: Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: leone.slavich@gmail.com

DOI: https://doi.org/10.1090/proc/13272
Received by editor(s): February 3, 2016
Received by editor(s) in revised form: February 18, 2016, and May 2, 2016
Published electronically: August 30, 2016
Communicated by: David Futer
Article copyright: © Copyright 2016 American Mathematical Society

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