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Would there even be any advantage to replacing the type II and type III Reidemeister moves with the following? Has this appeared in a publication?

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One thing lost is type II's ability to link completely unconnected strings, but Reidemeister moves aren't enough to axiomize classic knot theory anyway.

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    Could you make a clearer picture, since with the ascii symbols, you can't see which strand is over which strand. – 2017-02-22
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    This is an interesting idea, but your picture does not describe the move you have in mind, as Simon points out. If you draw it on paper and link to a camera image, we can try to edit that image in for you. – 2017-02-22
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    I made something. Thanks for editing for me. – 2017-02-23

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One thing you are always "allowed" to do in math is change the basic rules, and see what you get. A different example of what you are basically trying to do here is virtual knot theory, introduced by Kauffman. In virtual knot theory, they don't replace the three Reidemeister moves, but they add a new crossing type and a few moves that take it into account. This is a sub-field of knot theory, but has its own interesting quirks and theorems that many study without really looking at classical knot theory.

With that example in mind, what will (probably) happen here is you will find your move is not equivalent to the Type II and III moves. Thus, you will not have the classical knot theory, but a "new" knot theory.

  • What this means for knots: You will probably be able to find knots which are the unknot in classical knot theory, that are not the unknot in your knot theory. To do this would be a challenge! You will need to prove that the knot you have cannot be untangle with your two moves. In classical knot theory, this is usually done with invariants. So you will need to construct an invariant which can distinguish the unknot from your knot.

  • On the other hand: This may give you classical knot theory back. For that to be true, you will need to show that you can get the type II and III moves by only using type I and your move, and vice versa.

If you succeed in either of these, it would be a very nice paper and it sounds fun. Good luck!

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    With the exception of linking completely unconnected strings, type II and type III can be derived from this move using type I judiciously. Is the proof of such really worth a paper? – 2017-02-24
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    By itself, probably not. What would make it (possibly) worthy of a paper is some concrete examples of how this would differ from classical knot theory. Find a knot which can be unknoted in one but not in the other, or vice versa. – 2017-03-21
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    Like I said, it's not different. Just a smaller basis for the same theory. – 2017-03-22
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    Well, if there is some exception to when we can produce type II moves, then it is not the same theory. Thus, there are almost certainly going to be knots which are identical in classic knot theory which not the same in your theory. – 2017-03-22
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    Does type II allow you to connect strings/knots A and C where A is inside string/knot B and C is outside B? This planarity constraint is what I mean by the moves not being enough to axiomize knot theory. – 2017-03-22
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    I am not completely sure what you are trying to say here but I think the answer to your first question is No. Look [here](http://mathworld.wolfram.com/Tangle.html). You need to show that the tangles that represent type II and type III can be obtained from using only type I and your move (also in tangles inside of these tangles) to show your theory is the same. – 2017-03-22