Volume and Determinant Densities of Hyperbolic Rational Links

Adams, Colin; Calderon, Aaron J.; Jiang, Xinyi; Kästner, Alexander; Kehne, Gregory; Mayer, Nathaniel H.; Smith, Mia J.

doi:10.48550/arxiv.1510.06050

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2016

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“…Here v tet ≈ 1.01494 is the volume of a regular ideal tetrahedron. The upper bound in (2) shows that the volume density of any link is strictly less than v oct . Together with Conjecture 1.1, this implies:…”

Section: Introductionmentioning

confidence: 99%

“…For infinite sequences of alternating links without bigons, equation (2) implies that Spec vol restricted to such links lies in [v oct /2, v oct ].…”

Section: Introductionmentioning

confidence: 99%

“…al. [2] proved that Spec vol = [0, v oct ] and [0, v oct ] ⊂ Spec det , hence Spec vol ∩ Spec det = [0, v oct ]. Adams et.…”

Section: Introductionmentioning

confidence: 99%

“…al. [2] also showed that for any x ∈ [0, v oct ], there exists a sequence of knots K n with x as a common limit point of both the volume and determinant densities of K n .…”

Section: Introductionmentioning

confidence: 99%

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Geometrically and diagrammatically maximal knots

Champanerkar¹,

Kofman²,

Purcell³

2016

J. London Math. Soc.

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The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios.

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Section: Introductionmentioning

confidence: 99%

“…For infinite sequences of alternating links without bigons, equation (2) implies that Spec vol restricted to such links lies in [v oct /2, v oct ].…”

Section: Introductionmentioning

confidence: 99%

“…al. [2] proved that Spec vol = [0, v oct ] and [0, v oct ] ⊂ Spec det , hence Spec vol ∩ Spec det = [0, v oct ]. Adams et.…”

Section: Introductionmentioning

confidence: 99%

“…al. [2] also showed that for any x ∈ [0, v oct ], there exists a sequence of knots K n with x as a common limit point of both the volume and determinant densities of K n .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations