Calvo had the insight that understanding the deeper flype structure of prime, non-alternating diagrams led to greater efficiencies in his algorithm.
which comes from this potentially mindblowing paper by Meredith and Synder that I have just started reading, which faithfully embeds all knots (and knot equivalences) into the π-calculus (and weak bisimulations of processes).
... but it turns out flype is a real thing. I dare you to say Tait Flyping Conjecture with a straight face, though.
Arg, I can't understand why the encoding works. Even for Reidemeister-1 moves.
Suppose I compare
...B... ...|... A--|-\. ...\-/. ....... to ...B... ....\.. A--..|. ...\-/. .......
As portions of knots, these are identical: I just untwist the twisty bit to get from the first to the second. But, if I understand their encoding right, if I send a message from vertex A towards vertex B, it starves in the first diagram (because it's waiting for some "higher-priority" message to cross the overpass and write to the local channel u) but it makes its way to B in the second diagram.
What did I misunderstand? Are signals not as directed as I think they are? That was my main confusion at first, wondering whether a signal can wobble back and forth across an overpass building up a huge budget of u-writes there. But I think I've convinced myself everything is arranged so that a message is supposed to proceed around the knot in one direction once it's created.