Suppose Reidemeister 2 and 3 moves are valid on a knot/link diagram, but not 1. With the Whitney trick, you can annihilated a positive and a negative writhe. Picture But as you see, one writhe points up and the other down! I was not able to annihilate two opposite kinks with only R2/3 moves if both point up, and I doubt it can be done at all.
Can't it? (A possible proof would be giving a R matrix representation where kink-up and kink-down are different.)
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asked Oct 16, 2019 at 14:53
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1$\begingroup$ Forgetting the data of which strand crosses over which gives an immersion from $S^1$ to the plane (whose only singularities are isolated double points). Taking the unit tangent vector defines a map $S^1\to S^1$, and it's straightforward to show that its winding number doesn't change under isotopies or R2/R3 moves. Introducing two kinks which point up changes it by $2$, however. $\endgroup$– Bertram ArnoldOct 16, 2019 at 21:25
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$\begingroup$ Update: It seems my sidemark was correct - taking the R matrix of some direct sum of two Lie algebras (I tested A1+C) indeed up and down positive kink are different matrices. $\endgroup$– Hauke ReddmannOct 18, 2019 at 21:20
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