Phil Cook v2 wrote:

> Yay!

See, that's what I'm talkin bout! :-D

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

Yeah, but topology is friggin *weird*. Knot theory makes actual sense.

> 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.

See my other reply. I've got the algorithm wrong.

>> 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.

Probably. Actually, wait - there are only 2 crossings. No nontrivial 
knot has that few. Yes, it's definitely trivial. *sigh* Rusty...

>> 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.

Now, see, when I spend ages writing something like this, I kinda want 
people to go "hey, that's interesting. I had no idea this crap even 
existed!" But typically they go "OK, that's nice dear".

I just wish I could find a place where the stuff I know would actually 
impress people... *sigh*

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