  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.
