It passes now- thanks!!!

]]>Hello Dr. Moscovich,

I hope I’m responding to the right place.

I have attached the fixed html file that at least validates when uploaded here: http://validator.w3.org/ .

I also in the mean time found that asking my feed reader to execute the following script:

perl -pe ‘s/\x0c//; s/\x16//’

on the feed also made the feed behave with my feed reader. Specifically, one could, in a terminal:

perl -pe ‘s/\x0c//; s/\x16//’ ~/ldtopol_error.htm > ~/ldtopol_fixed_perl.htm

should output the fixed version in ldtopol_fixed_perl.htm, assuming ldtopol_error.htm is the current version in your home folder.

I have attached both versions, the former fixed by hand, and the latter fixed with the perl script. The problematic lines are all fixed & contained in the following too, for hopefully direct copy-paste:

In this paper we construct a combinatorial 1-cocycle for which is based on the HOMFLYPT invariant, see Theorem 4 in Section 11. It is called R(1) (“R” stands for Reidemeister). Fiedler then reproves Hatcher’s Theorem, for the figure eight knot, using only quantum topology. This is a major triumph! Quantum proofs of geometric-looking facts like that are rare and precious. It indicates to me that there is real gold in Fiedler’s approach. The conjecture is that the the crux of the proof works for any knot, leading to a new conjecture relating the quantum and geometric worlds. Namely:

Conjecture:Let be a long knot with non trivial Vassiliev invariant . Then R(1)(rot(K)-hat(K))=0 if and only if is a torus knot or a satellite with all pieces in the JSJ-decomposition of the complement are Seifert fibered.

The vague dream hiding in the wings is that there is a formula relating vales of to the hyperbolic volume of the knot complement. Even more vaguely, how much geometry of the knot complement does actually see, and how?

I have seen the above issue arise once before with wordpress, but I am not sure how it was fixed. Hopefully the plaintext should work, but if it doesn’t, just let me know, as I’m perfectly willing to try other things to help.

Thank you, -Michael

]]>Dear Michael,

Thank you very much for your comment! I only just now got around to seriously trying to address it- but I’m not really making any progress. In particular, deleting and reinserting spaces does not help, and neither does saving as unicode or as ascii. I wonder whether it might be an issue with wordpress? I really have no idea. As an experiment, if it would be OK, could you try to e-mail me the final few paragraphs, formatted by your computer? (might it be an issue with my machine?)

I’ll cut and paste, and this will give us some idea as to where the issue originates. Sorry for the trouble.

I have enjoyed following this blog for a couple years now, but I recently noticed that updates no longer appear in my rss reader. As per the following feed validator:

http://feedvalidator.org/check.cgi?url=http%3A%2F%2Fldtopology.wordpress.com%2Ffeed%2F ,

it appears that there are some stray characters that standard xml does not recognize; the Opera browser detects them when one clicks on the entries rss link, or one can use the validator here:

I have denoted the strange characters by “_*_”:

…Theorem 4 in Section 11. It is called _*_ R(1)

…title=’v_2(K)’ class=’latex’ />. Then _*_ R(1)(rot(K)-hat(K))=0 if and only if <i…

…ith all pieces in the JSJ-decomposition of the complement are Seifert _*_ fibered.

… is that there is a formula relating _*_ vales of <img src='http://s0.wp.com/lat…

Opera allows one to edit source files locally, and it appears that deleting the characters denoted above and reinserting a space there fixes the problem. One can check with http://validator.w3.org/ by uploading a local file with the specified changes. I'm sure there are other readers who would appreciate the change. I only became aware of your later posts, after checking the actual site.

Please let me know if there is any way I can help,

-Michael

Great point! From a topological perspective, you’d think that hat(K) should be defined for framed knots- but from a combinatorial perspective, fixing a blackboard framing (which you can modify by Reidemeister 1 moves) might perhaps be convenient… I look forward to peeking further into the paper!

]]>A general fact is that regardless of which of the above definitions of hat(K) you use, hat(K) and rot(K) are linearly dependent in the homology of the space of knots if and only if the knot K is a torus knot. edit: I suppose the advantage to using the blackboard framing for hat(K) is that you can view the evolution of the knot diagram as there being one fixed knot diagram on an S^2, and you simply slide the stereographic projection point along the knot diagram in that S^2. So the hat(K) class is sort of the most planar-looking among all possible normalizations.

]]>Dear Prof. Kalfagianni,

Thank you for this comment!

It’s only for the figure eight knot, as you point out (I’ve corrected the post)… how much more exciting it would be at “the next level” if there were a combinatorial proof of Hatcher’s theorem for all knots, and a direct explicit relation between quantum invariants associated to 1-cocycles in the space of knots, and geometry of the knot complement!

What are your thoughts about Fielder’s approach?

When you write:

“Fiedler then reproves Hatcher’s Theorem using only quantum topology. This is a major triumph! Quantum proofs of geometric-looking facts like that are rare and precious. It indicates to me that there is real gold in Fiedler’s approach.”

Do you mean that Fiedler’s paper contains a combinatorial proof of Hatcher’s theorem ? Or do you mean

that it contains a proof Hatcher’s theorem for the figure eight knot?