I can easily tell the mod p colorability of an alternating pretzel knot!

[onye]

A pretzel knot is called alternating if all the three twists have the
same sign. It is non-alternating if the three twists or tangles are
not of the same sign. For example, (1,2,1) and (1,-2,1) are
alternating and non-alternating respectively.

Now a significant part of my project is to stipulate an algorithm by
which we may look at a pretzel knot, and tell its mod p colorability.
I am also interested in an algorithm by which we may tell if a pretzel
knot is mod 3 or tri-colorable. This might prove to be quite tedious
but should be rewarding nonetheless.

But let me start with a general algorithm for the mod p colorability
of a pretzel knot. In knot theory, we define a tree as a graph that is
connected and has no closed cycles. Constructing a tree for a knot is
another way by which we may determine the determinant of a knot. There
is a beautiful theorem that says that for an alternating pretzel knot,
the number of maximal trees equals the determinant of the knot.

Remember that a knot is colorable mod p if the determinant of that
knot is divisible by p. For an alternating pretzel knot, the
relationship pq+pr+qr= the number of trees or determinant of the knot.

I know some of you are probably wondering if I could prove this
algorithm works in all cases as I have claimed. No worries, I am going
to set up the proof in my next posting. My only problem is that I will
need to draw pictures to illustrate my points. This is what I am
having difficulty doing.

For example, the (1,2,1) pretzel knot is another way of representing
the figure 8 knot. By the Alexander matrix of the figure 8 knot, we
already know that the determinant of a figure 8 knot is 5. Using the
notion of trees, we can also show that the maximum number of possible
trees in the figure 8 knot is 5. Therefore, we may conclude that the
figure 8 knot is colorable mod 5. Also, we can tell by this algorithm,
that the (3,2,2) pretzel knot is mod 12 colorable. Another question to
consider is if mod 12 colorability implies mod 6 colorability.

It is obvious from this algorithm, that the figure 8 knot is not tri-
colorable. Again, we can see this from the fact that
(1*2)+(2*1)+(1*1)=5 and 5 does not perfectly divide 3.

By this alogorithm, we may determine the colorability of all
alternating pretzel knots. Now, we must find a way to modify this rule
or algorithm to include all pretzel knots. This will underline a good
portion of the rest of my project.

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

Hi Onye,

This sounds like an interesting idea!

How exactly do you associate a tree to an alternating knot? What is a
"maximal tree"? How does one prove the theorem that the number of
maximal trees is the same as the (absolute value of?) the
determinant? Does the theorem apply to alternating knots that are not
pretzel? Why is the number of maximal trees for a pretzel knot equal
to pq+qr+pr? Is there a reference where I can read about this
theorem?

Can we also find a proof using the old ideas?

Dan

[Onyebuchi Samuel Okoro]

In my previous posting, I had indicated that a (3,2,2) pretzel knot was colorable mod 12. That is wrong. I guess I meant to push the 6 button instead of the 2. The (3,2,2) pretzel knot is actually colorable mod 16 i.e. (3*2)+(3*2)+(2*2)=16 But another question is if mod 16 colorability would imply mod 8 colorability, and if mod 8 colorability would imply mod 4 colorability and so forth. Of course we already know that no pretzel knot is colorable mod 2. I am still researching. I'll talk to you soon with my proof of how I came about the algorithm. But I would also have to explain the concept of trees so that my proof will make sense to you all. The concept of trees is something we did not talk about in class. At least I can't recall that we talk about it. But it is a beautiful concept nonetheless.

 

Thanks.

---

From: knotth...@googlegroups.com on behalf of onye
Sent: Mon 4/14/2008 11:23 AM
To: knot theory reu
Subject: I can easily tell the mod p colorability of an alternating pretzel knot!

[onye]

I am trying to find a proof using the old ideas also. Thanks.

> > portion of the rest of my project.- Hide quoted text -
>
> - Show quoted text -