Ball packings for links

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 R-3. (C) 2021 Elsevier Ltd. All rights reserved.