Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Started a page at link. More to add, especially some nice pictures!, but have to go to parents’ evening now.
I’m reading Milnor’s paper “Link Groups” so shall add stuff as I read it. This should also serve as warning to a certain Prof Porter (assuming it’s the same one!) that his 1980 paper is on my list of “things to read really soon”.
What paper is that? If you mean our Knots website. I will put up a link (pun not really intended but unavoidable). http://www.popmath.org.uk/exhib/knotexhib.html
There is also the excellent text book
N.D.Gilbert and T. Porter, Knots and Surfaces, textbook, Oxford University Press, November 1994.
and a version of the exhibition in booklet form (Mathematics and Knots, 1989). And a clutter of old boards with a physical copy of the knot exhibition which is somewhere in my ex-office in Bangor.
But I don’t seem to have a paper on knots from 1980, so perhaps, Andrew, its another Porter.
I may be able to add some pictures if I still have them on file somewhere, but I know someone has a file of them!
I checked the picture files are there as gifs so if you want to use any the usual process works. (Do please put in an acknowledgement if you do use some as the PopMath site likes that and it ain’t just Ronnie and Me. We also like the advertisement. )
That Porter is R. not T. so not me!
Porter, R. (1980), “Milnor’s μ-invariants and Massey products”, Transactions of the American Mathematical Society (American Mathematical Society) 257 (1): 39–71, doi:10.2307/1998124
May I ask why you want to put links up on the Lab, since then I may be able to add some compatible stuff in.
So now we have to be careful about writing like
this is how you create a [[link]]
? (-:
Tim, Bother! I suppose it was too good to be true that the author of a paper that I wanted to understand happened to be here.
Mike, ha-ha. Though perhaps I ought to go through the “linked from” pages and see if any of them shouldn’t point there.
I’ve added a couple more diagrams, and made the Milnor reference point to Math Reviews.
As to the reason, partly because there’s a computation that I want to do but to do it, I need to learn the basics of knots and links so I figured I ought to record those basics on the nLab as I learn them. Partly because knot theory has been an embarrassing gap in my topological knowledge for a while so I’m taking this opportunity to fill it.
Dick Crowell wrote a paper on link groups at about the same time as the following.
@article{crowell, Author = {Crowell, R. H.}, Journal = {Advances in Math.}, Pages = {210 - 238}, Title = {The derived module of a homomorphism}, Volume = {5}, Year = {1971}}
This paper does mention it I think, but he works with the Wirtinger presentation of the link group and examines the type of construction that leads to the Alexander polynomial from that viewpoint. I had a copy at one time but cannot think where it is or whether it has been shed due to lack of filing space!
BTW the original way of doing the Alexander poly (i.e. not via the know group and Fox derivatives) is fun if you have not seen it. (It is doable by UG students and allows for direct verification of invariance via Reidemeister moves.) (EDIT: It is fun even if you have seen it. :-))
Turns out I need to pay 30 dollars to see that article. Harumph.
Okay, here’s a question for you. I have a link, all the components are unknotted, and I want to compute something. It seems that the Milnor invariants are the right things to compute, are they? Also, what’s the simplest way to compute them (I have a drawing of said link, but the crossing number is high)? I found a paper on skein relations and Milnor invariants (technically, of the extension to strings).
I have just realised that Crowell’s paper is about the fundamental groups of complements of links not the version that Milnor considers (from what Wikipedia says).
I am no expert, BUT my supervisor from way back talked about Massey products being useful, see http://en.wikipedia.org/wiki/Massey_product
Searching on Massey products in Google threw up some more interesting results, e.g. http://www.jstor.org/pss/2001240
and more n-labish
http://arxiv.org/abs/0912.1775
but it looks as if Turaev and Stein and AMS Memoirs Derivatives of links: Milnor’s concordance invariants and …, Issues 425-427 By Tim D. Cochran may have summaries of Milnor’s results and ideas without having to pay $30
perhaps I ought to go through the “linked from” pages and see if any of them shouldn’t point there
Done; you’re fine.
Toby asked (at link):
Now that [MO] said [the Hopf link is Brunnian], should we ask whether the Hopf link is considered to be a Borromean link? It seems to me that it should be (and in fact so should the unknot and the empty link).
A quick google search doesn’t come up with a definition of “a Borromean link” as anything other than another name for the Borromean rings. Can you explain what you meant?
I misread
It is possible to link together circles in such a way that removing any one makes the others fall apart.
as a description of a class of links called ‘Borromean’. To avoid more such confusion, I will change ‘Borromean links’ to ‘Borromean link’ and ‘Borromean rings’. (Three rings, but one link.)
I have added linking number, but for some unknown reason, I cannot get arrows with the svg-editor and my standard easy xypic code produces nothing in codecogs. Heigh ho! C’est la vie. :-(
Can you add the Xy code for the diagram that you want (either here or there), so somebody else can give it a try?
Thanks, I will try on a dry run i.e. in another ’document’ and then see if further help is needed.The diagrams are simply the two oriented overpass diagrams! I tried to describe them underneath the place for the diagram on linking number.
I had a go. I hope that I got them the right way around!
The arrows are done by “markers”. When you draw or select a line in the SVG-editor then there is a menu option for adding markers at the start, middle, or end of the line. These are little drop-down menus and are labelled “s”, “m”, “e”. Confusingly, once you select an arrow-head it disappears from the list, but it’s still there in that it still renders and the blank space in the list where it was can be used to select it again. (I’ve emailed Jacques about that).
Yes. I found the arrows very confusing to add and the help on them not that helpful.
The diagrams are fine. Great thanks.
I hope to add stuff on HOMFLY etc, that I know before getting onto stuff that I know less well. It would be great to link it all into the monoidal categories and other aspects, but I am not sure I will have time. (<- I have been officially retired but … . Note the tense of the verb.)
I have been officially retired but … . Note the tense of the verb.
I understand. I have relatives in academia for whom “officially retired” is the best description!
Yippee. A bit of fiddling, plus Googling for the bug that was stopping me starting Inkscape and I (i) have got Inkscape working and (ii) have done the first Reidemeister move.
By the way, my meaning was also that ’I was retired’. I did not choose the state except under pressure. I have managed to enjoy doing lots of research since that time, and have been lucky in getting visits to various places, but the work I do now is unpaid (i.e. lots of refereeing, reviewing, editing etc.) as well as the enjoyable bits trying to extend the n-Lab, write the Menagerie, complete my tentative study of HQFts etc. I do not teach at Bangor at all at the moment.
1 to 20 of 20