Densities of hyperbolic cusp invariants of knots and links
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- by Colin Adams, Rose Kaplan-Kelly, Michael Moore, Brandon Shapiro, Shruthi Sridhar and Joshua Wakefield PDF
- Proc. Amer. Math. Soc. 146 (2018), 4073-4089 Request permission
Abstract:
We find that cusp densities of hyperbolic knots in $S^3$ include a dense subset of $[0,0.6826\dots ]$ and those of links are a dense subset of $[0,0.853\dots ]$. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing density for links is bounded above by $3.1263\dots$. Moreover, there is a sequence of links with cusp crossing density approaching 3. For two-component hyperbolic links, cusp crossing density is shown to include a dense subset of the interval $[0,1.6923\dots ]$ and for all hyperbolic links, cusp crossing density is shown to include a dense subset of $[0, 2.120\dots ]$.References
- Colin C. Adams, Thrice-punctured spheres in hyperbolic $3$-manifolds, Trans. Amer. Math. Soc. 287 (1985), no. 2, 645–656. MR 768730, DOI 10.1090/S0002-9947-1985-0768730-6
- Colin C. Adams, Augmented alternating link complements are hyperbolic, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 115–130. MR 903861
- Colin Adams, Cusp densities of hyperbolic 3-manifolds, Proc. Edinb. Math. Soc. (2) 45 (2002), no. 2, 277–284. MR 1912639, DOI 10.1017/S0013091501000025
- C. Adams, A. Colestock, J. Fowler, W. Gillam, and E. Katerman, Cusp size bounds from singular surfaces in hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 358 (2006), no. 2, 727–741. MR 2177038, DOI 10.1090/S0002-9947-05-03662-7
- Colin Adams, Bipyramids and bounds on volumes of hyperbolic links, Topology Appl. 222 (2017), 100–114. MR 3630197, DOI 10.1016/j.topol.2017.03.002
- Colin Adams, Alexander Kastner, Aaron Calderon, Xinyi Jiang, Gregory Kehne, Nathaniel Mayer, and Mia Smith, Volume and determinant densities of hyperbolic rational links, J. Knot Theory Ramifications 26 (2017), no. 1, 1750002, 13. MR 3597249, DOI 10.1142/S021821651750002X
- K. Böröczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978), no. 3-4, 243–261. MR 512399, DOI 10.1007/BF01902361
- Stephan D. Burton, The spectra of volume and determinant densities of links, Topology Appl. 211 (2016), 38–55. MR 3545288, DOI 10.1016/j.topol.2016.08.022
- Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell, Volume bounds for weaving knots, Algebr. Geom. Topol. 16 (2016), no. 6, 3301–3323. MR 3584259, DOI 10.2140/agt.2016.16.3301
- Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell, Density spectra for knots, J. Knot Theory Ramifications 25 (2016), no. 3, 1640001, 11. MR 3475068, DOI 10.1142/S0218216516400010
- Marc Culler, Nathan M Dunfield, Matthias Goerner, and Jeffrey R Weeks, SnapPy a computer program for studying the geometry and topology of $3$-manifolds http://snappy.computop.org.
- Mario Eudave-Muñoz and J. Luecke, Knots with bounded cusp volume yet large tunnel number, J. Knot Theory Ramifications 8 (1999), no. 4, 437–446. MR 1697382, DOI 10.1142/S0218216599000304
- Robert Thijs Kozma and Jenő Szirmai, Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types, Monatsh. Math. 168 (2012), no. 1, 27–47. MR 2971738, DOI 10.1007/s00605-012-0393-x
- Robert Meyerhoff, Sphere-packing and volume in hyperbolic $3$-space, Comment. Math. Helv. 61 (1986), no. 2, 271–278. MR 856090, DOI 10.1007/BF02621915
- Carlo Petronio and Michele Tocchet, Notes on the peripheral volume of hyperbolic 3-manifolds, Math. Nachr. 287 (2014), no. 5-6, 677–685. MR 3193943, DOI 10.1002/mana.201200180
- Alan W. Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991), no. 1, 171–184. MR 1099096, DOI 10.1112/jlms/s2-43.1.171
- W. Thurston, The geometry and topology of 3-manifolds: Princeton University Notes, 1980, http://library.msri.org/books/gt3m/PDF/.
Additional Information
- Colin Adams
- Affiliation: Department of Mathematics and Statistics, Bascom Hall, 33 Stetson Court, Williams College, Williamstown, Massachusetts 01267
- MR Author ID: 22975
- Email: colin.c.adams@williams.edu
- Rose Kaplan-Kelly
- Affiliation: Department of Mathematics, Temple University, Wachman Hall, 1805 North Broad Street, Philadelphia, Pennsylvania 19122
- Email: rose.kaplan-kelly@temple.edu
- Michael Moore
- Affiliation: 2600 Netherland Avenue, Apt. 2120, Bronx, New York 10463
- Email: mrm2231@columbia.edu
- Brandon Shapiro
- Affiliation: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853
- Email: bts82@cornell.edu
- Shruthi Sridhar
- Affiliation: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853
- Address at time of publication: Fine Hall, Washington Road, Princeton, New Jersey 08544
- Email: ssridhar@princeton.edu
- Joshua Wakefield
- Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
- Address at time of publication: Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Email: joshpw@mit.edu
- Received by editor(s): July 5, 2017
- Received by editor(s) in revised form: November 27, 2017
- Published electronically: May 24, 2018
- Additional Notes: This research was supported in part by NSF grant DMS-1347804.
- Communicated by: Kenneth Bromberg
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4073-4089
- MSC (2010): Primary 57M50
- DOI: https://doi.org/10.1090/proc/14068
- MathSciNet review: 3825860