Ball packings for links

TL;DR

This paper establishes an upper bound on the minimum number of solid balls needed to realize a link in 3D space, relating it to the link's crossing number, and provides an explicit construction algorithm.

Contribution

It introduces a bound of five times the crossing number for ball packings of links and develops an explicit algorithm for constructing such packings using Lorentz geometry.

Findings

Proves that ball(L) ≤ 5 * cr(L) for links.

Uses Lorentz geometry and circle packing theorems in the proof.

Provides an explicit construction algorithm for link representations.

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 $ball(L)\leq 5 cr(L)$ where $cr(L)$ denotes the crossing number of $L$. To this end, we use Lorentz geometry applied to ball packings. The well-known Koebe-Andreev-Thurston circle packing Theorem is also an important brick for the proof. Our approach yields to an algorithm to construct explicitly the desired necklace representation of $L$ in the 3-dimensional space.