Equivalent Knot Projections

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

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