Ball packings for links

Jorge Luis Ramírez Alfonsín, Ivan Rasskin

Published 2020-10-01Version 1

The ball number of a link $L$, denoted by $ball(L)$, is the minimum number of solid balls needed to realize a necklace representing $L$. In this paper, we show that $ball(L)\leq 5 cr(L)$ where $cr(L)$ denotes the crossing number of $L$. To this end, we use Lorenz geometry applied to ball packings. Our approach yields to an algorithm to construct explicitly the desired necklace representation of $L$ in the 3-dimensional space.