Discussion on a-note-on-notation

[marg...@math.utah.edu]

Hi Tim,

If these two notations are really the same, can you give an explicit
function that takes as its input the Dowker notation, and outputs the
crossing permutation, and vice versa?

Dan

[quiddit...@gmail.com]

Generating the Dowker notation from the crossing permutation (the cp)
is really easy.

If the cp does not start with a negative label then rewrite the cp so
that it does. (this corresponds to starting the cp at an
undercrossing)
The even odd pairs of numbers for each crossing generated from the
algorithm for the Dowker notation can be gotten by looking at the
position in the cp of the negative and positive label for that
crossing. Dowker notation is all positive only for alternating
diagrams but giving the appropriate sign changes for non-alternating
graphs is easy using the crossing permutation. Since the crossing
permutation tells you which crossings are over and under crossings you
can just change sign appropriately for deviations from alternation.
(the way this works out in the end is if the label in the cp at an
even position is negative or a the label at an odd position in the cp
is positive then flip the sign of the associated number in the Dowker
notation, usually the dowker notation contains just the even or just
the odd numbers ordered according to the order of their paired number
so just flipping the sign of whichever one you are using will do the
trick).

Generating the crossing permutation from the Dowker notation can be
done in much the same manner but in reverse. Take the even odd pairs
one at a time and put label 1 and -1 in the position of the crossing
permutation which corresponds to the even and odd numbers, putting
them in the right place according to the sign of the numbers in the
Dowker notation. and there you have it.

[marg...@math.utah.edu]

Hi Tim,

Can you post the picture from the last slide in your talk and tell us
what it means?

Dan