Knot Tying Schema

I’ve noticed that whenever my two-year-old wants to tie a knot, she takes the two strands that she’s interested in and repeatedly twists them around each other.  Of course this doesn’t produce any kind of knot at all (or I should say it only produces the unknot).  But this twisting appears to be the only basic move that she knows (and she learned it herself) for producing knots.

It made me wonder what the basic moves are that we use to tie knots.  Every knot theorist knows that there are three basic moves for transforming one picture of a mathematical knot into another (possibly quite different looking) picture of the same knot.  These moves are called the Reidemeister moves.  Any two pictures of the same knot can be made to look like each other using just these moves (and “plane isotopies”).  Similarly it seems to me that mathematically you really only need two moves to tie any knot, or what I mean to say is to go from a picture of a straight string to a picture of any knot.  Here is an illustration of those two moves:

basic_moves

Of course after doing several of these moves, you will want to glue the two ends of the string together if you wnat to get a picture of a mathematical knot.

When people tie knots, it does seem like there is some finite set of moves they use, but the catalogue of moves seems to be bigger than these two moves I illustrated above.  For instance there are moves like “wrap one end around a loop” or “put one end through a hole” or “follow one strand through a whole series of moves.”  Of course each of these can be broken up into a sequence of the two basic moves, but it is not always useful to break things down into the simplest possible moves in practice.  It makes me wonder what the catalogue of moves is that people use.

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