Ball packings for links

Abstract

The ball number of a link L, denoted by ball(L), is the minimum number of solid balls (not necessarily of the same size) needed to realize a necklace representing L. In this paper, we show that ball(L)≤5cr(L) where cr(L) denotes the crossing number of a nontrivial nonsplittable link L. To this end, we use the connection of the Lorentz geometry with the ball packings. The well-known Koebe–Andreev–Thurston circle packing Theorem is also an important brick for the proof. Our approach yields an algorithm to construct explicitly the desired necklace representation of L in R3.