This inspired the conjecture that non-planar twistor diagrams correspond to non-planar Feynman diagrams and a concrete proposal for realizing the earlier proposal that the contribution of non-planar diagrams could be calculated by transforming them to planar ones by using the procedure applied in knot theories to eliminate crossings by reducing the knot diagram with crossing to a combination of two diagrams for which the crossing is replaced with reconnection. The Wikipedia article about magnetic reconnection explains what reconnection means. More explicitly, the two reconnections for crossing line pair (AB,CD) correspond to the non-crossing line pairs (AD,BC) and (AC,BD).
I do not bother to type the 5 pages of text here. Instead I give a link to the article Still about non-planar twistor diagrams at my homepage. For background see the chapter Generalized Feynman Diagrams as Generalized Braids of "Towards M-matrix".