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Ball packings for links

Jorge Luis Ramírez Alfonsín, Ivan Rasskin

TL;DR

This paper shows that the ball number of a link $L$ is the minimum number of solid balls needed to realize a necklace representing £L, where cr(L) denotes the crossing number of $L$.

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.

Ball packings for links

TL;DR

This paper shows that the ball number of a link is the minimum number of solid balls needed to realize a necklace representing £L, where cr(L) denotes the crossing number of .

Abstract

The ball number of a link , denoted by , is the minimum number of solid balls (not necessarily of the same size) needed to realize a necklace representing . In this paper, we show that where denotes the crossing number of . 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 in the 3-dimensional space.

Paper Structure

This paper contains 14 sections, 8 theorems, 26 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The space of $d$-balls.
  3. From spherical caps to $d$-balls
  4. The intersection angle of two $d$-balls
  5. The hyperbolic model of $\mathsf{Balls}(\widehat{\mathbb{R}^d})$
  6. The Lorentzian model of $\mathsf{Balls}(\widehat{\mathbb{R}^d})$
  7. The Möbius group
  8. $d$-ball packings
  9. Pyramidal disk systems
  10. The crossing ball system
  11. The proof of the main theorem.
  12. From links to disk packable graphs.
  13. Proof of Theorem \ref{['theorem']}
  14. The necklace algorithm

Key Result

Theorem 1

Let $L$ be a link. Then,

Figures (18)

  • Figure 1: (Left) A knot diagram of the trefoil (denoted by $3_1$): the simplest non-trivial knot. (Right) A link diagram of the Hopf link (denoted by $2_1^2$) : the simplest non-trivial link.
  • Figure 2: A necklace representation of the trefoil.
  • Figure 3: The intersection angle of two disks.
  • Figure 4: Geometric interpretation of the isomorphisms of Equation (\ref{['eq:isos2']}).
  • Figure 5: A 2-ball packing with its carrier.
  • ...and 13 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Conjecture 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Theorem 2
  • Proposition 3.1
  • proof
  • Lemma 2
  • ...and 8 more