## Identifying Equivalent Knot Projections

As discussed in a previous page, each knot projection may possess up
to 4N notations, where N is the crossing number. (There is just one exception,
the trivial projection with 0 crossings has 1 > 4 x 0 notations) In order to
switch from one notation to the other, one moves the starting point and/or the
orientation. These notations are related as follows.
a) Orientation the same: (i,j) -> (c+i,c+j) where c is constant for all pairs.
If c+i and/or c+j is larger than 2N, one subtracts 2N.

b) Orientation inversed: (i,j) -> (c-i,c-j) where c is constant for all pairs.
If c-i and/or c-j is less than 1, one adds 2N.

Therefore, given a (drawable) notation, one may easily obtain all other
notations yielding the same projection. Once a criterion is established that
will keep exactly one such notation and reject the others, the classification
of regular knot projections has been completed.

There is a number of such criteria that one may use. One may impose a **total
ordering** relation to the notations, and among notations yielding
identical projections, keep the one that is ordered ahead of all the others.
Each such relation will lead to a distinct criterion.

A brief footnote here: by *total ordering* relation of a set S one
means a subset O of S x S such that:

- a) s in S => (s,s) in O,
- b) (s,t) in O and
(t,s) in O => s=t,
- c)(r,s) and (s,t) in O => (r,t) in O, and
- d) for all s,t in S,
either (s,t) or (t,s) in O.

To see the criterion chosen for this project and to proceed with the rest of
the discussion, click **here**.

*Charilaos Aneziris, charilaos_aneziris@standardandpoors.com*

**Copyright 1995**

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