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• CommentRowNumber1.
• CommentAuthorTim_Porter
• CommentTimeOct 28th 2010

I have started a page on knot groups. So far I have outlined how to get the Dehn presentation.

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeOct 29th 2010
• (edited Oct 29th 2010)

Completed how to get the Dehn presentation and have given a worked example of a cinquefoil knot.

One point that intrigues me is how can one categorify the Tietze transformation technology. It seems difficult even at the second level.. i.e. a presentation of 2-group. Has any one seen anything like this?

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeOct 31st 2010

No, but I’m intrigued! At times(*) I have wondered (and actually worked on) how to define a fundamental crossed module of some sort of cellular space (say CW complex, or $|delta$-complex) as a slim and easy counterpart to the big and ugly fundamental bigroupoid that I know it has. When I think about trying these ideas on a knot-complement, then I think I run into some trouble due to a lack of cellular structure. If there was a nice algorithm to decompose the knot-complement, I might be able to do this. Otherwise, some wort of movie moves approach might work…

(*)when I was supposed to be finishing my thesis write-up

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeOct 31st 2010

Joao has written a paper The Fundamental Crossed Module of the Complement of a Knotted Surface. Transactions of the American Mathematical Society. 361 (2009), 4593-4630. which does something a bit on those lines.

In fact one of the reasons for writing these draft pages on knot groups etc. was, for me, to revisit some of the web of ideas van Kampen, Wirtinger, covering spaces, Tietze transformations, etc. to see if the current knowledge on vKTs etc in higher dimensions could allow the development of 2-dimensional analogues of hypothetical missing pieces.

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeOct 31st 2010

Joao’s publications are listed at

http://ferrari.dmat.fct.unl.pt/~jnm/

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeOct 31st 2010

Thanks, Tim. I’ll check them out.