  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 $x^4 + y^4 + z^4 = 2t^2$.
