We derive bounds on the length of the meridian and the cusp volume of 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 with meridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.

It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite volume hyperbolic 3-manifold. In this paper, we address the generalization of this question to hyperbolic 3-manifolds admitting tunnel systems. We show that there exist finite volume hyperbolic 3-manifolds with a single cusp, with a system of n tunnels, n − 1 of which come arbitrarily close to self-intersecting. This gives evidence that systems of unknotting tunnels may not be isotopic to geodesics in tunnel number n manifolds. In order to show this result, we prove there is a geometrically finite hyperbolic structure on a (1; n)-compression body with a system of n core tunnels, n − 1 of which self-intersect.

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 non-alternating links with volume density approaching v 8 , where v 8 is the volume of the 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 [2π log det(K)]/c(K). We prove that the closure of the set of determinant densities contains the set [0, v 8 ].

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