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
- Proc. Amer. Math. Soc. 146 (2018), 4073-4089
- DOI: https://doi.org/10.1090/proc/14068
- Published electronically: May 24, 2018
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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
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Bibliographic 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