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.
Created isotopy and circle, also a bit of housekeeping (adding redirects and drop-downs) at knot and knot invariants.
For circle, my thought was to present it as an example of … just about everything! But I’m sure that there’s things I’ve missed, so the intention is that it not be a boring page “the circle is the units in ” but rather “the circle is an example of all these different things”.
(On that thought, I’ve sometimes wondered how much of the undergraduate syllabus could be obtained by applying the centipede principle to .)
I have commented a bit more on isotopy of knots. We probably need a page on ambient isotopy as the present definition can go a bit strange in the generality it is given. (You need smoothness or PL, or use ambient isotopy, otherwise all knots are istopic to the unknot!)
I also added two references. :-) (<- sales are flagging!!)
Didn’t know that! I added the definition of “ambient isotopy” to the page isotopy - separating the two didn’t seem particularly worthwhile.
Fine. Would you like something on Reidemeister moves, then we could handle regular isotopy as well. I can provide some pictures!
Of course! The more the merrier. Perhaps it’s worth having a “Knot and Link Theory - Contents” page listing all the topics so that it’s easy to see where the blanks are. Ben put a few at knot invariants.
I have added [bridge number]], but still need to do pictures.
@Toby Toby asked
Why not 0? (I don’t see any bridges in the obvious diagram.) Are there numerical operations on bridge numbers that only work for the unknot if its bridge number is 1? Can we motivate the existence of a bridge in the unknot by proper application of negative thinking?
Put one twist in the unknot and you get one bridge. That is the unsatisfactory answer!
A more ’correct’ answer would have been clear to you if I had continued the entry last night! If we take the sum of two knots (first tie one then the other on the rope) then . This can be shown using elementary arguments for non-trivial knots, but of course it would fai,l for the trivial case were either an unknot, if we had , as K+unknot is K up to equivalence, but setting is then fine.
I suspect that there is a DEEP reason as well. I recall that the coefficients of the Alexander polynomial have geometric interpretations, and suspect that one is related to the bridge number. There are also deep definitions of the bridge number as something to do with the number of components in the tangle sum decomposition of the pair , but that was found by google just now and I do not undrstand the terms (also do not have access to the journal (as usual)).
(Edit: I found the following that is nice:
http://www.warwick.ac.uk/~maaac/lecture13.html
Andrew: you may like that as it is smooth<- in sense of almost synonym for ’cool’.
That is on Brian Sanderson’s website and he has a lot of links to knots. :-))
Tim, incidentally, when you cut-and-paste URLs, if you put them in angle brackets then they get converted to clickable links and the link is shown as the text: http://www.warwick.ac.uk/~maaac/lecture13.html. This works both here and on the nLab.
@Andrew. Thanks. I had ’forgotten’ that or something like that. As you can see I have made the necessary change. :-)
Right, I’ve created knot theory - contents and included it on the page knot. As someone edits each of the pages in this area we should add the contents to it. The template is:
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Contents### {: .clickToReveal}
###Contents### {: .clickToHide tabindex="0"}
+--{: .hide}
[[!include knot theory - contents]]
=--
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
(some pages already have a drop-down list, that should be removed first)
Clearly the contents needs expanding!
Also, given the vast array of reference material on this, it might be nice to have one page wherein it is all gathered (and classified). This would be in addition to page-specific references.
I gave up trying to draw the two forms of trefoil using svg. I tried to upload the picture as just that, a picture but that went wrong, so I uploaded it as a pdf file which can be looked at but is not seen on the page. Suggestions?
My main suggestion is to wait until I’ve finished preparing Monday’s lecture and then I’ll have a go at drawing it! The only reason I didn’t earlier when I saw the page on bridge number (hmm, nearly missed the ’g’ out of ’bridge’; that would be a very different invariant!) was because I don’t know which trefoil is which! Now that you’ve uploaded a PDF, I can take a look and draw the right one.
Were you trying the online SVG editor? It can be a little tricky to get used to (particularly as one cannot ’right click’, which I find is my instinct in many circumstances - I expect to get a useful context menu), though I like the fact that it integrates well with instiki and is particularly good for diagrams including maths. But how you prepare the SVG doesn’t matter, you can do it with inkscape if you like. The only thing to be ware of is that you need to have “plain svg” and you need to strip off the first line (the <?xml line). Then I think that simply cut-and-pasting the SVG will work.
I have tried to get Inkscape working on my MacBook, but so far without success! The thing I found difficult was the click to turn to curve. …. It disappeared as soon as I tried to use it! I lost all my Corel-draw diagrams from the book some years ago when changing machines. I could no longer open them as Corel had scrapped backwards convertibility. (Does xfig produce svg files?)
I would not worry about which is which of the trefoils as the bridge number is mirror invariant.
later on I will try another of Ben’s list, but also need Knot group, Wirtinger presentation, Reidemeister moves, knot diagram …. It will be a useful displacement activity whilst my (so-called) brain considers other things :-)
I’ve had a go at converting the diagrams to SVGs using the inline SVG editor. They perhaps aren’t the prettiest possible, but they’re much better than my first attempt! As they’re now displayed on the page, I shifted the PDF file to the bottom of the page with the acknowledgements. I hope that the wording is okay.
@Andrew. I like them. Can there be a bit more separation between them and the trefoil could be a bit less big…. .
Thanks to a bit of CSS wizardry (which took longer than just simply editing the picture would have done), the trefoil is now scaled to the same height as the other one and a gap is forced between them. This does mean that the line thicknesses differ …
That looks great.
1 to 18 of 18