Comments on A Neighborhood of Infinity: Untangling with Continued Fractions: Part 2

How the fraction will happen if we are using 3 str...

2008-08-27T05:43:00.000-07:00

How the fraction will happen if we are using 3 strings?

Finally, everything makes sense!
And I didn't h...

2008-08-25T01:53:00.000-07:00

Finally, everything makes sense!
And I didn't have to read more than what you wrote.

The definition of 0 (in K) was so unsurprising, and my misguided intuition told me that directions weren't important, that when thinking about it and when eventually doing real physical experiments, I was starting with S0 instead of 0 without noticing.

Now I can see STS0=0, and with a bit more effort TST0=0.
Using correct definitions really helps...
Sorry for the confusion.

I'm still wondering whether the cyclic group generated by STS is the whole stabilizer of 0.

sigh I've been repressing my fascination with kn...

2008-08-24T19:48:00.000-07:00

*sigh* I've been repressing my fascination with knot theory since I was introduced to the concept way back in college. But your articles have forced me now to purchase Formal Knot Theory, by Kauffman. The thought of the applications of this theory are bewitching! But that's just drop of the ocean of pleasure to be had at cocktail parties:

Jrandom: "So, what do you do?"
Me: "Oh, I'm a Knot theorist by profession. How about you?"
Jrandom: ... response silenced by sounds of choking on the salmon mousse canapé.

You also mentioned a few posts ago that you have a LaTeX to Blog/HTML system you built. Would you be willing to share the source code? Or, to make a literate Haskell source entry of it on your blog?

Thanks for your explanation.
I'm afraid I didn'...

2008-08-24T18:07:00.000-07:00

Thanks for your explanation.
I'm afraid I didn't read your first comment carefully enough.

I was still expecting G and K to be essentially the same (as moves on rubik's cube and positions on it), and was doubting that q was 1-1.
Now I realize that in your first comment, when you wrote STS0=0, you meant the 0 in K, not k (where it acts on via p). (Of course it doesn't matter much when q is 1-1.)

My intuition doesn't work here, so I guess I should follow the references and take a look at the exact definitions.

Call the group of twists and rotations G. Call the...

2008-08-24T15:32:00.000-07:00

Call the group of twists and rotations G. Call the group generated by s and t, g. There is an isomorphism, call it p:G->g. Call the set of rational tangles (up to isotopy) K. Call the extended rationals k. There is a 1-1 correspondence between these sets; call it q:K->k.

Not only is p a homomorphism between groups but the action of the groups on their respective sets is compatible with p and q.

So if x is in G then not only is p(xy)=p(x)p(y) etc. we also have that for any rational tangle t in K, q(xt)=p(x)q(t).

There is also a map G->K taking x to x0. Every rational tangle is of the form x0 for some x in G so this map is surjective. But it's not an isomorphism. Maybe when I said "Once we know the sequence..." I gave the impression that it was a unique sequence, but I was just using 'the' to refer to the composition in the previous sentence. The extended rational however is unique, there is precisely one for each rational tangle even though there are multiple ways to build this rational using s and t from 0.

I'm probably completely missing what you're saying, but the above should disambiguate a lot.

I do claim that the sequence of s and t's should take you all the way back to precisely 0.

It's the paragraph
So, given any [only n...

2008-08-24T15:17:00.000-07:00

It's the paragraph
So, given any [only now that I'm citing it I see that you have that little word twice] rational tangle we simply need to find the corresponding extended rational and then figure out how to write it as a composition of s's and t's applied to 0. [...] And once we know the sequence, we can then apply it in reverse to untangle the tangle.

This could only work completely if there were a 1-1 correspondence. Maybe the trick is that it works good enough, i.e. gives an easily untangled tangle.

I'm not sure what I've misled you to believe. I'm ...

2008-08-24T14:27:00.000-07:00

I'm not sure what I've misled you to believe. I'm not claiming a 1-1 correspondence between the elements of our group and the rational tangles so it's fine for STS0=0 (and sts0=0).

The plan to figure out how to get the extended rat...

2008-08-24T01:48:00.000-07:00

The plan to figure out how to get the extended rational by applying s and t to 0 seems to imply that the stabilizer of 0 is trivial. This is not the case, STS also stabilizes 0. There must be something you're still hiding from us. Maybe the whole stabilizer is the group generated by STS, so the plan gives a way to go to a boring tangle with extended rational 0 which is easy to untangle.
Anyway, fascinating stuff, I'm curious how the story continues.

Comments on A Neighborhood of Infinity: Untangling with Continued Fractions: Part 2

How the fraction will happen if we are using 3 str...

Finally, everything makes sense!And I didn't h...

*sigh* I've been repressing my fascination with kn...

Thanks for your explanation.I'm afraid I didn'...