Discrete geometry and topology of entanglement of straight lines in 3-space

Abstract

We propose an unexpected twist to description of the geometry and topology of configurations of n straight lines considered as a whole 3D entity (because the lines are inseparably linked pairwise while having linking numbers 1/2 or -1/2) and named n-cross. Our theory stems from our work on configurations of mutually touching straight cylinders but, along with the previously introduced Ring matrix (that controls the encaging of each line by other lines), we now introduce fundamental direction 3D matrices (whose entries 0, 1, and -1 are signs of mixed products of line orientation vector triples). Discrete motion/connection combination principle established in the space of Ring and direction matrices (forming a groupoid and resembling moves in Loyd 15-puzzle game or Khovanov homology) allows one to discern topologically different configurations of lines with elementary methods and without link diagrams of knot theory. However, with the help of so-called projection 3D matrix we also integrated our matrix approach into the knot theory and established topological invariants for line entanglement in both approaches thus connecting 2D projections with 3D configurations. With Jones polynomials we show that an n-cross is a link of pairwise connected n unknots in a topological sense. The known results of the knot theory for rigid isotopy of 6 and 7 lines are reproduced and a novel result for 8 lines is given. With our approach we reach nuances of the geometry of lines never investigated before. It may find applications in Algebra, Discrete Geometry and Topology, and Quantum Physics.