## 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