Volumes, determinants, and meridian lengths of hyperbolic links
We study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of nonalternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi log det(K)/c(K). We prove that the closure of the set of determinant densities contains the set [0, v_8]. We examine the conjecture, due to Champanerkar, Kofman, and Purcell that vol(K) < 2 pi log det (K) for alternating hyperbolic links, where vol(K) = vol(S^3\ K) is the hyperbolic volume and det(K) is the determinant of K. We prove that the conjecture holds for 2bridge links, alternating 3braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.We derive bounds on the length of the meridian and the cusp volumeof hyperbolic knots in terms of the topology of essential surfaces spanned by the knot.We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound.As applications we find knot diagrammatic upper bounds on the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots withmeridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.
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Electronic Theses & Dissertations
 Copyright Status
 Attribution 4.0 International
 Material Type

Theses
 Authors

Burton, Stephan D., 1987
 Thesis Advisors

Kalfagianni, Efstratia
 Committee Members

Bell, Robert
Hedden, Matthew
Schmidt, Benjamin
 Date
 2017
 Program of Study

Mathematics  Doctor of Philosophy
 Degree Level

Doctoral
 Language

English
 Pages
 viii, 135 pages
 ISBN

9781369725087
1369725086
 Permalink
 https://doi.org/doi:10.25335/2rasym18