Links in orthoplicial Apollonian packings

Jorge L. Ramírez Alfonsín, Iván Rasskin

Published 2023-01-08Version 1

In this paper, we introduce a connection between Apollonian packings and links. We present new representations of links embedded in the tangency graph of orthoplicial Apollonian packings and show that any algebraic link can be projected onto the tangency graph of a cubic Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links, to reinterpret the correspondence of rational tangles and rational numbers, and to find primitive solutions of the Diophantine equation $x^4 + y^4 + z^4 = 2t^2$.