And lo On Tue, 17 Feb 2009 11:45:40 -0000, Invisible <voi### [at] devnull> did  
spake thusly:

> There now follows a large brain dump concerning knot theory...

Yay!

> At one time, it was briefly theorised that maybe the ~105 elements of  
> the periodic table were each a little tangle of energy, and different  
> kinds of tangling gave rise to different chemical properties.
>
> (With String Theory, the idea seems to be coming back somewhat. But I  
> digress...)

Yeah interesting how some ideas can come back from the grave.

> So what actually *is* knot theory? Well, it's the study of mathematical  
> "knots". As you might expect, these abstract entities have properties  
> similar to but not quite the same as a real knot in a piece of string.

Also perhaps worth pointing out it's only one area of study in topology.

<snip>

> One rather entertaining way goes something like this. (I've probably  
> screwed up the algorithm; this is from memory.)
>
> - Pick a starting point on the string, and draw an arrow representing a  
> direction. Doesn't matter what you pick, but stick to it.
>
> - Trace your way around the knot. Each time you reach a crossing, number  
> it, starting from 1. If the strand you're on goes over the top, use a  
> positive number. If it goes under, assign a negative number.

Okay let's try the trefoil knot.

> - Write down a list of all the pairs of numbers at each crossing.

-1, 4
2, -5
-3, 6

> - Throw away the lowest number in each pair (ignoring sign).

Leaves 4,-5,6. Hmmm? Okay let's try that again following Dowker notation.

1,4
2,5
3,6

As this is an alternating knot, no changes in signs required.

Write out the odd numbers with corresponding entry beneath

1, 3, 5
4, 6, 2

Throw away the top numbers to leave 4,6,2.

> An alternative way to describe knots is by "braid theory".
>
> A "braid" is a series of vertical strands. Initially, they are all  
> parallel. If you say "+3", that means that strand 3 and strand 4 swap  
> places, with strand 3 going over the top of strand 4. Alternatively,  
> "-3" means the same swap, but strand 4 going over the top.
>
> In this way, you can say "-3, +5, +2". This describes a sequence of  
> strand swaps, starting from the top and working downwards. Something  
> like this:
>
>    1   2   3   4   5   6
>    |   |   |   |   |   |
>    |   |    \ /    |   |
>    |   |     /     |   |
>    |   |    / \    |   |
>    |   |   |   |   |   |
>    |   |   |   |    \ /
>    |   |   |   |     \
>    |   |   |   |    / \
>    |   |   |   |   |   |
>    |    \ /    |   |   |
>    |     \     |   |   |
>    |    / \    |   |   |
>    |   |   |   |   |   |
>    1   2   3   4   5   6
>
> So that's a braid. Now if you imagine taking this and bending it over so  
> that the ends at the top connect with the ends at the bottom, this would  
> make a closed loop. In fact, in this case, the result would be *several*  
> closed loops. The 1 strand would be an unknot, not connected to anything  
> else. Strands 5 and 6 would become a single strand, which can then be  
> untwizzled to make an unknot. And strands 2, 3 and 4 would be connected;  
> off the top of my head, I'm not sure if this would be a nontrivial knot.

Trivial, It's a rubber-band twisted twice.

> This has been another broadcast brought to you by an under-employed  
> computer science graduate, for the benefit of similarly over-interested  
> souls. TTFN!

Interesting, polish it up and stick it on your blog.

-- 
Phil Cook

--
I once tried to be apathetic, but I just couldn't be bothered
http://flipc.blogspot.com

Post a reply to this message