## 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*

**Copyright 1995**

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