View this page "Imaginary Knot Theory"

[bch...@comcast.net]

Click on http://groups.google.com/group/knottheoryreu/web/imaginary-knot-theory?hl=en
- or copy & paste it into your browser's address bar if that doesn't
work.

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

Hi Brian,

Sounds like a good idea. Can you tell us exactly what an imaginary
knot is? How do you create two knots out of one? Perhaps some
pictures are in order.

Why are physicists interested in imaginary knots?

[quiddit...@gmail.com]

Such a powerful new move makes it seem like it might be hard to come
up with two knots that we know are different in imaginary knot theory.
What are examples of two imaginary knots which are different? Do most
of the knots we have come to know and love stay unique in this world
or is the trefoil the same as the figure eight etc.

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

Excellent question, Tim. Can we prove or disprove that the trefoil
knot is equivalent to the unknot under these moves, for instance?
That might be a good first question.

Also, can you show us why the 1-unknot is equivalent to the 2-unknot?

Eventually, it might be good to see if we can find any kinds of
invariants of these knots, such as colorability.

Dan

[bch...@comcast.net]

The book talks about how using a slide equivalance, Korvanov? or
someone was able to build up a knot theory based off of the braid
group. This formulation went well with something called coupling
theory, which seems to be related to quantum mechanics. I'm still not
sure what that is exactly, but I'm currently reading a bit about it.

[Jason]

if you read on pg. 4 in the article I posted it mentions that the
hecke algebra is based off of a unit q. reading on the internet has
made implications that these algebras are what you are talking about
through something called q-mathematics. search me for what these are.
But Vaughan Jones was the one given credit for making a theory of
knots on the braid group.

[Charlotte]

I don't see how the new move would change a knot or link that is
already prime (because we can't do R. moves on a prime knot anyway,
right?) unless after you break up an intersection by sliding an arc
away from it, you can 'attatch' an arc from another broken
intersection. Is that possible? What becomes of the 'loose ends'?

[bch...@comcast.net]

What I'm looking at must then be a bit different because it mentions
in the book that the invention of imaginary knot theory in braid form
was independently invented in braid form by a M. Khovanov. So I guess
what it is saying is that he built it up from the ground up using the
new "Reidemeister" moves.

[bch...@comcast.net]

Sorry this is a duplicate, but I didn't realize that by pushing reply
to author, it only sent it to that person. I figured it would send it
in the same body as the original text. So here it is: I am under the
impression that you can slide one arc and reconnect it to another one
as long as it doesn't "jump" off the arc onto another one that crosses
it. So, it will be interesting to look at examples because the book
shows an illustration where an overcrossing it split into two
seperated arcs. I'll add some more pictures and examples of moves that
are possible this weekend for those interested. It gave me some
insight as to why the braid group and specifically permutations would
be helpful. I want to look at how keeping track of how the arcs
disassemble, whether that keeps that information of the knot intact
enough to either tell what it is, or to tell whether they are the same/
different. I hope this makes sense, and I'll add pictures to help make
it clearer.

[bch...@comcast.net]

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

Hi Brian,

Can you explain your picture that transforms the trefoil into the
link? What moves are you using?

Is this a familiar link?

Dan

[bch...@comcast.net]

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

Hi Brian,

Nice pictures!

What move are you using to get from Step 3 to Step 4? I don't see it.

Can we prove that you can't get a type I move from the imaginary
moves? In particular, can we show that the knot diagram with 1
crossing is not equivalent to the diagram with 0 crossings? Are there
any diagrams we can show to be inequivalent to the diagram with 0
crossings?

I am a little unclear--do we get the Reidemeister type 3 move or not?
You show us the type 2 move, and you seem to indicate that type 3 is
possible, but I don't see the appropriate picture.

Dan

[bch...@comcast.net]

I havn't been able to prove that you can't get the type 1 move from
IKT(imaginary knot theory), but I am thinking that we are going to get
the answer they are NOT equivalent.

[bch...@comcast.net]

Also going to step 4 all I am doing is a type 2 R. move followed by a
type 3 move(both together). So just seperating the two arcs and then
erasing the extra arc. I'm going to post it in a second, but it seems
weird that I can't find a way to replicate the type 1 R. move of
traditional knot theory because I have found plenty of cases where I
can add a twist. If I can add a twist, then it seems like I should be
able to remove one. I'm going to try and really go slow and see what I
can come up with. Maybe I am just missing something.