Densities of Hyperbolic Cusp Invariants (original) (raw)

Densities of hyperbolic cusp invariants of knots and links

Proceedings of the American Mathematical Society

We find that cusp densities of hyperbolic knots in S 3 include a dense subset of [0, 0.6826. .. ] and those of links are a dense subset of [0, 0.853. .. ]. 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. .. . 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. .. ] and for all hyperbolic links, cusp crossing density is shown to include a dense subset of [0, 2.120. .. ].

Hyperbolic Invariants of Knots and Links

Transactions of the American Mathematical Society, 1991

Tables of values for the hyperbolic volume, number of symmetries, cusp volume and conformai invariants of the cusps are given for hyperbolic knots through ten crossings and hyperbolic links of 2, 3 and 4 components through 9 crossings. The horoball patterns and the canonical triangulations are displayed for knots through eight crossings and for particularly interesting additional examples of knots and links.

Volume and determinant densities of hyperbolic rational links

Journal of Knot Theory and Its Ramifications, 2017

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [Formula: see text]. We construct sequences of alternating knots whose volume and determinant densities both converge to any [Formula: see text]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants.

Cusp size bounds from singular surfaces in hyperbolic 3-manifolds

Transactions of the American Mathematical Society, 2005

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, ℓ \ell -curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, ℓ \ell -curve length and maximal cusp volume for hyperbolic knots in S 3 \mathbb {S}^3 depending on crossing number. Particular improved bounds are obtained for alternating knots.

THE NEXT SIMPLEST HYPERBOLIC KNOTS

Journal of Knot Theory and Its Ramifications, 2004

We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these "simple" hyperbolic knots have high crossing number. We also compute their Jones polynomials.

Turaev hyperbolicity of classical and virtual knots

Algebraic & Geometric Topology, 2021

By work of W. Thurston, knots and links in the 3-sphere are known to either be torus links, or to contain an essential torus in their complement, or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction. In the case of a classical link, we can also define the classical Turaev volume, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction for the classical projections only. We then investigate these new invariants.

Hyperbolic knots are not generic

arXiv: Geometric Topology, 2019

We show that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer crossings does not converge to 111 as nnn approaches infinity. Moreover, we show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of the prime knots of nnn or fewer crossings does not converge to 000 as nnn approaches infinity.

Waist size for cusps in hyperbolic 3-manifolds II

Geometriae Dedicata, 2019

The waist size of a cusp in an orientable hyperbolic 3manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In this paper, it is proved that the next two smallest waist sizes are realized uniquely for the cusps in the 52 knot complement and the manifold obtained by (2,1)-surgery on the Whitehead link. One application is an improvement on the universal upper bound for the length of an unknotting tunnel in a 2-cusped hyperbolic 3-manifold.

On the Question of Genericity of Hyperbolic Knots

International Mathematics Research Notices

A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer crossings approaches 111 as nnn approaches infinity. In this article, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.