Discussion on research-questions

[bch...@comcast.net]

Do we know for sure that there exists two equivalent diagrams with n
crossings that are not trivially the same(under elementary
deformations, rotations, or reflections-I call these trivial because
they don't require reidemeister moves to deform them into another
equivalent diagram. Maybe looking at an example would help start you.

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

Hi Miles,

Great ideas (in your pdf, under Files)!

You are right that there are many more 2-crossing diagrams than we
thought! Which of these become equivalent on the 2-sphere? What
about if we allow ourselves to flip the plane over? Is there any
other sense in which some of the diagrams are morally the same?

Do you have a conjecture for the relationship between the number of
crossings and the number of bounded regions? I am not completely sure
I know what you mean by "bounded region"--aren't there bounded regions
in your picture that are not shaded?

I like the circle pictures. Now, the set of symmetries of a circle
diagram forms a group (but what do we mean by symmetries?). Can you
relate the number of knot diagrams to the number of circle diagrams,
together with the order of this symmetry group? It seems like that is
what you are getting at. Is there a nice formula?

Do the circle diagrams help us to algorithmically list all diagrams?

Dan

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

I am confused. I just wrote a 5 paragraph reply, and I only see the
first 2 here. Am I doing something wrong?

Dan

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

Miles,

Your weave picture is very intriguing. This is worth thinking about
more (it's a big open question!). Do you have a guess for what is the
smallest number of extra crossings needed?

Dan

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

Here is (hopefully) the rest of that message:

[miles.fore]

I have been thinking about many of the questions you asked. If you
consider the diagrams to be on a sphere (I believe you called it a
stereographic projection, or something similar) then many of them
become equivalent. I haven't yet tried to find out which are
equivalent, or if there is a way to find it in the general case. I do
find it noteworthy that if one considers diagrams on a sphere then my
weave diagram example (from my other pdf file) doesn't work because
the two diagrams are the same. Assuming the weave example was a
counterexample (which I don't know if it is) it raises the question of
if there is a counterexample that uses diagrams on a sphere.

I don't yet have a conjecture for the relationship between the number
of bounded regions and the number of crossings. This is something
that I am trying to figure out. My current ideas on how to get this is
to first come up with an algorithm to write down all of these, and
then derive from this the number of bounded region diagrams given
there are a certain number of regions.
On a side note, the bounded regions are probably just what you think
they are, I just haven't defined them very well. Where you say there
are some bounded regions that aren't shaded it is probably the white
regions. I considered white a shading as I didn't want to try and
shade three different levels. So in each picture there are three
colors, and the "outside" part of the diagram is not a bounded region.

As for the last questions, it seems like there might be a way to
relate the number of diagrams to the number of bounded regions, and
the order of the symmetry of the group. That is what I am trying to
find, but I haven't yet. If it does work out there might be a nice
formula for the number of diagrams with n crossings. Also (if the
above questions can be answered) it seems that there might be an
algorithm given for drawing all diagrams. Last night I tried listing
all circle diagrams for a three crossing diagram, but became too tired
to finish. I'll have to try again today.

> Dn

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

Miles,

I looked at your "Circle diagrams" and your "Upper bound..." files.
These are very interesting.

Can anyone see a way to get the exact number of 3-crossing diagrams
from Miles's count for the number of circle pictures? What about n-
crossing diagrams?

Is there a more efficient way to get from one diagram to the other in
the weave pictures? If not, then this would answer Open Question #1
in The Knot Book. Anyone?

Dan