Links in orthoplicial Apollonian packings

TL;DR

This paper connects Apollonian packings with knot theory, introducing new link representations, realizing algebraic links in orthoplicial packings, and improving bounds on algebraic link complexity.

Contribution

It establishes a novel link between Apollonian packings and knot theory, providing new representations and geometric insights into algebraic links and related Diophantine equations.

Findings

Any algebraic link can be realized in the cubic section of orthoplicial Apollonian packing.

Improved upper bounds on the ball number of certain algebraic links.

Reinterpretation of rational tangles and solutions to a specific Diophantine equation.

Abstract

In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation $x^4 + y^4 + z^4 = 2t^2$.