Fig. 10: Conservation of the colored invariant Q_gray in topologically allowed strand crossings.

By rotating the picture if necessary, one can assume that the black color of the bicoloring is on the right hand side when moving upwards along either of the strands of the crossing. The pictures (a), (b) and (c) depict the cases that can occur. In (a) and (b) the strands partaking in the crossing change are parts of the same loop L_i; note that the self-writhes of L_i and \({L}_{i}^{{\prime} }\) differ by ±2. The difference between the two cases is that in (a), the quaternion q₁ anticommutes with j, because the path corresponding to q₁ passes under an odd number of red and blue strands in total, whereas, for analogous reasons, in (b), q₁ commutes with j. The crossing change in (a) and (b) does not alter the linking number of L_i with other gray loops, so Q_gray remains unchanged. In (c), the strands are part of loops L_i and L_j. Note that \(\alpha ({L}_{i}^{{\prime} })\alpha ({L}_{j}^{{\prime} })=-\alpha ({L}_{i})\alpha ({L}_{j})\), but the introduced sign is canceled as the total modulo-two linking number between gray loops is altered by 1, so Q_gray remains unchanged.