Denoting Regular Knot Projections, the Sequel

Answers to questions raised in the previous page

1) No, the notation is not unique. One may see that by altering the starting point and/or the orientation, one may obtain a different notation. For instance in the example shown here, if one were to start from the right of point A instead of the left, the notation would be {(1,6),(3,8),(5,2),(7,4)} instead of {(1,4),(3,6),(5,8),(7,2)}. In general an N crossing projection possesses (at most) 4N notations, 2N for each orientation. (In the example provided here, there are only 2 notations instead of 8 due to symmetries).

2) No, in fact when the number of crossings reaches infinity, the probability that there is a projection for some notation, tends to 0. We shall leave this discussion for later. Now we shall only mention that if a notation is drawable, and if the corresponding knot shadow is a connected sum of k prime shadows, there are (at most) 2^k possible projections. For more on this subject, click here.

3) It depends on the kind of the Reidemeister move.

First move: a pair (i,i+1) or (i+1,i) is added or removed, while all numbers higher than i increase or decrease by 2.
Second move: two pairs (i,j),(i+1,j+1) or (i,j+1),(i+1,j) are added or removed; numbers between i and j are increased or decreased by 2, numbers higher than j increase or decrease by 4.
Third move: three pairs (i,j),(i',k),(j',k'), where |i'-i|=|j'-j|=|k'-k|=1, are replaced by (i,k'),(i',j'),(j,k).

4) It is possible to apply color tests and to obtain the Alexander polynomials from this notation; in order to obtain the more recently invented knot characteristics one would first have to obtain the braid whose closure is the denoted knot projection.

Charilaos Aneziris, charilaos_aneziris@standardandpoors.com

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