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:

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

Charilaos Aneziris,

Copyright 1995


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