Volume and determinant densities of hyperbolic rational links

Abstract

Abstract. 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 [0, voct]. We construct sequences of alternating knots whose volume and determinant densities both converge to any x ∈ [0, voct]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants.

Geometrically and diagrammatically maximal knots

“…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:…”

“…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 ].…”

“…Adams et 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 .…”

“…Ideas for the proof of Theorem 1.4 are due to Agol. His unpublished results were mentioned in [17], but the geometric argument suggested in [17] for the lower bound is flawed because it is based on existing volume bounds in [3,21], and these bounds only imply that for K n as above, the asymptotic volume density lies in [v oct /2, v oct ] (see equation (2)). The proof of geometric maximality, particularly the asymptotically correct lower volume bounds, uses a combination of two main ideas.…”

The Determinant and Volume of 2-Bridge Links and Alternating 3-Braids

“…To obtain upper bounds on the volumes of knots we largely rely on work of Adams [1] who gave an upper bound in terms of volumes of bipyramids. Adams et al [2] used this upper bound to study the volume densities of 2-bridge knots. Other useful upper bounds in the case of highly twisted knots are due to Lackenby, Agol and Thurston [11] and Futer, Kalfagianni and Purcell [8].…”