Comments on A Neighborhood of Infinity: Untangling with Continued Fractions: Part 1

Leon,

In answer to (3), what we really want...

2008-08-23T14:57:00.000-07:00

Leon,

In answer to (3), what we really want are, I think, linear names.

Nice. Although, one thing that strikes me about ...

2008-08-23T13:07:00.000-07:00

Nice. Although, one thing that strikes me about your monadic abstraction is the naming issue, specifically, names need to be referred to on the R.H.S. of a function once, and only once. So it begs the questions:

1. Is there a reasonable or useful interpretation when a name is referred to more than once?

2. Similarly, what if a name is introduced but not referred to at all?

3. Is there an alternate abstraction that enforces this condition?

From reading the comments here, I suspect the answer to #3 is "Conway Notation." Great post, I look forward to reading the rest!

leithaus,

Only by name.

2008-08-23T07:38:00.000-07:00

leithaus,

Only by name.

Do you know Kassel's Quantum Groups book?

2008-08-22T23:24:00.000-07:00

Do you know Kassel's Quantum Groups book?

leithaus,

Almost everything I know about kn...

2008-08-22T15:57:00.000-07:00

leithaus,

Almost everything I know about knots comes from Kauffman's book "Knots and Physics" (and countless mostly-forgotten seminars from my student days). I don't recall anything about a relationship between Hopf algebras and quandles, at least not a direct one.

So... here's a question for you. How do quandles and Hopf algebras relate to each other? i know you can get a quandle out of a group with conjugation, but i'm fuzzy on the Hopf-crossed widget connection.

Your readers may be interested in the application ...

2008-08-21T09:37:00.000-07:00

Your readers may be interested in the application of rational tangles to biology. For instance the article Modeling protein-DNA complexes with tangles (PDF) is a fun introduction.

leithaus,

I can't talk about Conway notatio...

2008-08-20T16:53:00.000-07:00

leithaus,

I can't talk about Conway notation until I've talked about the classification of rational tangles. Or at least talked a bit more about rational tangles.

i think you should at least make a nod to Conway's...

2008-08-20T16:38:00.000-07:00

i think you should at least make a nod to Conway's knotation. If you stack cup and cap; and you place them (suitably rotated) side by side, then together with the two crossings you get Conway's knotation combinators. You can get a good description from Rob Scharein's thesis. For that matter, knotplot is a cool site and a cool tool.

Nice teaser.. :)

2008-08-17T17:43:00.000-07:00

Nice teaser.. :)

Your first example of a block:
(b,c) <- bloc...

2008-08-17T04:52:00.000-07:00

Your first example of a block:
(b,c) <- block (a,b)
seems to have a typo. I think this should be:
(c,d) <- block (a,b)

Small typo, I think: the first example should be <...

2008-08-17T00:17:00.000-07:00

Small typo, I think: the first example should be (c,d) <- block (a,b).

Comments on A Neighborhood of Infinity: Untangling with Continued Fractions: Part 1

Leon,In answer to (3), what we really want...

Nice. Although, one thing that strikes me about ...

leithaus,Only by name.

Do you know Kassel's Quantum Groups book?

leithaus,Almost everything I know about kn...

So... here's a question for you. How do quandles and Hopf algebras relate to each other? i know you can get a quandle out of a group with conjugation, but i'm fuzzy on the Hopf-crossed widget connection.

Your readers may be interested in the application ...

leithaus,I can't talk about Conway notatio...

i think you should at least make a nod to Conway's...

Nice teaser.. :)

Your first example of a block:(b,c) <- bloc...

Small typo, I think: the first example should be <...

Leon,

In answer to (3), what we really want...

leithaus,

Only by name.

leithaus,

Almost everything I know about kn...

leithaus,

I can't talk about Conway notatio...

Your first example of a block:
(b,c) <- bloc...