Fig. 11: Conservation of the colored invariant Qgray in topologically allowed reconnections.
a, b The cases to consider. The difference between them is the total number of times Li crosses under a red or a blue strand, modulo 2, when traversing from the point x to y. a In the case of an even number, the loop Li splits into two loops \({L}_{i}^{{\prime} }\) and \({L}_{i}^{{\prime\prime} }\). b In the case of an odd number, the quaternions qi anticommute with j and \({q}_{i}^{-1}=-{q}_{1}\), which are important facts used to deduce the conservation of the invariant Qgray.
