On the hull numbers of torus links

Introduced in [1] by Jason Cantarella, Greg Kuperberg, Robert B. Kusner, and John M. Sullivan, the hull number is a complexity measure for links similar to the bridge number. By definition, it is the maximum integer n such that for any realization of the link there is a point c ∈ R such that any plane through c intersects the curves representing the link at least 2n times. Clearly, the hull number is less than or equal to the bridge number. An elegant statement proved in [1] says that the hull number of a nontrivial knot is at least 2. By an integral-geometric argument this yields a new proof of Fáry-Milnor theorem. The authors posed the question of finding links and knots with large hull numbers. Unexpectedly enough, it is difficult already to find a knot with hull number at least 3. It was conjectured that the (3, 4)-torus knot has hull number 3. However, we show this to be false (see Figure 1). In the present paper we study hull numbers of torus links and prove two lower bounds. First, for non-trivial torus links with p components (and more generally, for any link with p pairwise non-trivially linked components) we show that the hull number is at least 3 5 p. This bound is sharp in the sense that for any p there are p-component links with components non-trivially linked and with hull number ⌈ 5 p⌉. In particular, these are (p, p)-torus links (“multi-Hopf links”). Secondly, for the hull number of a (p, q)-torus link with p ≤ q we prove the lower bound 1 2 p. In both cases, the scheme of the proof is to show that a plane intersecting the link in a small number of points cuts off an accordingly small portion of the link, and to apply Helly’s theorem. The idea to use Helly’s theorem for the study of hull numbers appeared already in [1].