Discussion on knot-calculator

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

Tim,

This sounds like a great idea for a project. Can you explain your
"crossing permutation notation"? It looks like the diagram in your
PowerPoint presentation (under Files) is a good example. How does
this notation relate to other notations, such as the Dowker notation?

What about pretzel knots?

Can you have your program check for colorability? Group
colorability? It might be interesting to program the multiplication
in the symmetric group. We may want to look into existing group
theory programs, like GAP, or MAGMA.

Prof. M

[quiddit...@gmail.com]

The Crossing permutation notation is produced simply by labeling all
the crossings of a knot diagram with the integers from 1 to n and then
traversing the knot in one direction or the other writing out a string
of numbers as you go. You write the positive label of a traversed
crossing if you are on the over crossing or the negative of the label
if you are going along the undercrossing. As a convention I always
start my strings with the overcrossing of the 1 crossing. The
crossing permutation of the trefoil is "1, -2, 3, -1, 2, -3"

I don't yet fully understand the properties of the crossing
permutation. The crossing permutation notation is not sufficient to
completely uniquely determine a knot like I had originally thought, it
is unable for instance to distinguish between a right handed and left
handed trefoil. I have created a more complicated notation which I
call cpr (crossing permutation region) notation which I believe is
sufficient to uniquely determine a knot diagram but I do not yet have
a proof. Also the CPR notation is considerably bulkier and more
difficult to deal with than the crossing permutation. I believe the
specification of the handedness of the crossings might be sufficient
to resolve the ambiguities in the crossing permutation notation but I
am not certain.

However the crossing permutation does carry all necessary information
to calculate the Alexander polynomial of the knot. So the crossing
permutation is sufficient to distinguish any two knots which can be
distinguished by their Alexander polynomials and if crossing
handedness is added then more is possible (since for instance then you
can tell apart the right and left handed trefoil)

As far as the relation to the Dowker notation or any other notations I
don't know. The dowker notation certainly seems related and is
possibly equivalent to the crossing permutation since they seem to
encode very similar information. The Dowker notation is more compact
than the cp but I think some transparency as to its meaning is lost.

I'm planning on including all of this in a more detailed write up
which I will upload to the site once I get it done, or at least get it
started.

As for pretzel knots I was planning on also putting an option to be
able to input any pretzel knot and have the Alexander polynomial
generated as well. For the moment my program is going to base all its
calculations on the crossing permutation information and I have worked
out the crossing permutation for the (2, n) torus knot which is simply
"1, -2, .... , n, -1, 2, .... , -n" I haven't worked out the cp for
the general pretzel knot and I was simply planning on reaching the
stage of generating correct Alexander polynomials for the (2, n) torus
knots and any manually entered knots that I can think of and then
begin take the program from there.

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

Tim,

A good project for Monday might be to give us pseudocode for what you
plan to program. I am very interested in your plan of attack.

Dan

[miles.fore]

This may be a silly question, but how do we know that crossing
permutation notation uniquely defines a knot? It seems like it
should, but could it be the case that two knots which aren't
equivalent could somehow have the same crossing permutation notation
(for some diagrams of the knots)?

[Alex]

Is there any way to look at your code if we don't have c#?