On isotopic weavings (Q1423838)

A weaving \(W=({\mathcal L}, P, m)\) consists of a finite set \(\mathcal L\) of pairwise intersecting straight lines in real affine plane such that no three of these lines have a common intersection point, the set \(P\) of all intersection points of any two lines of \(\mathcal L\), and a map \(m : {\mathcal L} \times {\mathcal L} \setminus \{(k,k)\mid k \in {\mathcal L}\} \to \{-1,1\}\) with \(m(k,l) = -m(l,k)\). If \(m(l,k) = 1\), the line \(l\) is called to be locally above the line \(k\) at their intersection point \(l \wedge k \in P\). A weaving \(W\) is called realizable if there is a configuration \(\mathcal C\) of straight lines in real affine 3-space which can be projected orthogonally onto \({\mathcal L}\) via some projection \(\rho\) such that dist\(( \rho^{-1}(p) \cap l^*, p) > \text{ dist}( \rho^{-1}(p) \cap k^*, p)\) for any two lines \(l,k\) of \({\mathcal L}\) with \(p=l\wedge k\) and \(m(l,k) =1\), where \(k^*,l^* \in {\mathcal C}\) with \(\rho(k^*) = k\) and \(\rho(l^*) = l\). Two weavings \(W\) and \(W'\) are said to be isotopic if there is an isotopy of the plane moving the lines of \(W\) to the lines of \(W'\) such that all lines of \(W\) remain straight lines during this isotopy. The author proves that there exist isotopic weavings one of which is realizable and the other is not. This is achieved by giving a necessary and sufficient condition for a weaving isotopic to some special weaving defined by the author to be realizable.