WEAVING PATTERNS OF LINES AND LINE SEGMENTS IN SPACE

A weaving W is a simple arrangement of lines (or line segments) in the plane together with a binary relation specifying which line is ''above'' the other. A system of lines (or line segments) in 3-space is called a realization of W, if its projection into the plane is Wand the ''above-below'' relations between the lines respect the specifications. Two weavings are equivalent if the underlying arrangements of lines are combinatorially equivalent and the ''above below'' relations are the same. An equivalence class of weavings is said to be a weaving pattern. A weaving pattern is realizable if at least one element of the equivalence class has a three-dimensional realization. A weaving (pattern) W is called perfect if, along each line (line segment) of W, the lines intersecting it are alternately ''above'' and ''below.'' We prove that (i) a perfect weaving pattern of n lines is realizable if and only if n less-than-or-equal-to 3, (ii) a perfect m by n weaving pattern of line segments (in a grid-like fashion) is realizable if and only if min(m, n) less-than-or-equal-to 3, (iii) if n is sufficiently large, then almost all weaving patterns of n lines are nonrealizable.