Geometrically and diagrammatically maximal knots

Abstract

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.

26 pages, 12 figures. V2: Updated references, removed hypothesis from Theorem 1.4, added Corollary 1.11, and made other minor wording changes. V3: Discussion on weaving knots has been moved into another paper. V4: Added Figures 10 and 12, minor wording changes

References in corpus (3)

• Volume bounds for weaving knots
• Hyperbolic semi-adequate links
• Approximations to the volume of hyperbolic knots

Cited by in corpus (18)

• Volume bounds for weaving knots
• Geometry of biperiodic alternating links
• Universal Knot Diagrams
• Right-angled polyhedra and alternating links
• Spanning Trees and Mahler Measure
• Mahler Measure and the Vol-Det Conjecture
• Determinant density and biperiodic alternating links
• Alternating links on surfaces and volume bounds
• Fully Augmented Links in the Thickened Torus
• Density spectra for knots
• A survey of hyperbolic knot theory
• The Determinant and Volume of 2-Bridge Links and Alternating 3-Braids
• Stable presentation length of 3-manifold groups
• A Quantum Invariant of Links in $T^2\times I$ with Volume Conjecture Behavior
• Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles
• Examples of homological torsion and volume growth
• Generalized bipyramids and hyperbolic volumes of tiling links
• Graph Complexity and Link Colorings