Low Dimensional Topology

April 6, 2013

New connection between geometric and quantum realms

Filed under: Hyperbolic geometry,Knot theory,Quantum topology — dmoskovich @ 9:41 am

A paper by Thomas Fiedler has just appeared on arXiv, describing a new link between geometric and quantum topology of knots. http://arxiv.org/abs/1304.0970

This is big news!!

In the geometric approach, a knot is viewed as its complement- a 3-manifold with boundary, which you might imagine as what is left after a small worm eats out a knot-shaped path of that big apple that is S^3. This complement can be equipped with a geometric structure (an instance of geometrization), and we can measure invariants such as its hyperbolic volume. You might fancy thinking of the geometric approach as a knot theorist’s “general relativity”.

In the quantum approach, on the other hand, a knot is viewed as its diagram, that is a configuration of crossings (minute, subatomic if you will, neighbourhoods in which “something happens”), and is understood by how subsets of those crossings connect together combinatorially. So you might think of the quantum approach as a knot theorist’s “quantum mechanics”.

Like general relativity and quantum mechanics in Physics, geometric and quantum topology of knots are both very useful, but the relationship between them remains a mystery. Do we get geometry if we zoom out of a knot diagram (I certainly hope so!). If so, how, and (more metaphysically), why?

There are a number of conjectures which relate geometric and quantum topology. Most famous are the Volume Conjecture, the AJ conjecture, and the Asymptotic Expansion Conjecture. The “story” for these conjectures seems to factor through physics, and they seem a bit technical and unenlightening to me- I’ve never succeeded in getting excited about any of these conjectures, despite having tried.

Fiedler’s approach, on the other hand, looks great! Quantum invariants can be viewed as combinatorial 0-cocycles in a moduli space of knots. What happens if we try to compute higher cocycles in the same (or a closely related) space? 1-cocycles, say? This idea would look natural to a smooth topologist or to a hard-core algebraic topologist, but it seems to have been off the radar of the quantum topological community- bravo Fiedler!

I defer to Fiedler’s introduction:

The connection to geometry is based on a result of Hatcher. It is well known that the classification problem for knots is equivalent to the classification problem of long knots, i.e. a smoothly embedded arcs in 3-space which go to infinity outside a compact set as a straight line. Let K be a long knot and let M_K be its topological moduli space. There are two natural loops in M_K: the rotation of K around the long axis, which we call \mathrm{rot}(K) and another loop, which we call Hatchers loop \mathrm{hat}(K) (compare [17]). It is defined as follows: one puts a pearl (i.e. a small 3-ball B) on the (closure of the) knot K in the 3-sphere. The part of K in S^3\setminus B is a long knot. Pushing B once along the knot induces Hatcher’s loop in M_K. The following theorem is an immediate consequence of a result of Hatcher (see [17]).

Theorem 1 (Hatcher)
Let K be a long knot which is not a satellite (i.e. there is no incompressible torus in its complement in the 3-sphere). Then \mathrm{rot}(K) and \mathrm{hat}(K) represent the same class in H_1(M_K;Z) if and only if K is a torus knot.

The hard direction of this theorem is of course to prove that for a hyperbolic knot these two loops are not homologous. This follows from the minimal models for the topological moduli spaces of hyperbolic and of torus knots which were constructed by Hatcher (see [8] and [7] for the case of satellites). Hatchers construction is mainly based on very deep results in 3-dimensional topology: the Smith conjecture, the Smale conjecture, the Linearization conjecture (which is a consequence of the spherical case in Perelmann’s work) and the result of Gordon-Luecke that a knot in the 3-sphere is determined by its complement.

In this paper we construct a combinatorial 1-cocycle for M_K 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 K be a long knot with non trivial Vassiliev invariant v_2(K). Then R(1)(rot(K)-hat(K))=0 if and only if K 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?

R(1) seems genuinely new, deep, and worthy of substantial further study. This is tremendously exciting!

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8 Comments »

  1. Dear Dr Moskovich:

    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?

    Comment by Effie Kalfagianni — April 7, 2013 @ 10:03 am | Reply

    • 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?

      Comment by dmoskovich — April 7, 2013 @ 6:45 pm | Reply

  2. I’ve only started peeking at the paper. The class Fiedler calls hat(K), at least in the arXiv v1 version of the paper is only well-defined up to addition of a multiple of the rot(K) class. The idea is you slide a ball along the knot, which you view as a framing of the tangent space of the knot with an extension to a framing of the tangent space of S^3. That bead you can think of as determining an element of SO(4), whereby multiplication by the inverse will translate the knot so that the bead is now in a standard configuration, centred at (1,0,0,0), and the knot’s tangent space there is in the direction of the vector i. You then do stereographic projection. So when sliding the bead along the knot the vectors normal to the knot in the bead’s framing are not uniquely determined. You could use the homologically-trivial framing (standard longitude) but that is not what Fiedler is using. He is using the blackboard framing (to use Rolfsen-style terminology).

    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.

    Comment by Ryan Budney — April 7, 2013 @ 10:06 pm | Reply

    • 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!

      Comment by dmoskovich — April 7, 2013 @ 10:59 pm | Reply

  3. Hello Dr. Moscovich,

    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:

    http://validator.w3.org/check?uri=http%3A%2F%2Fldtopology.wordpress.com%2Ffeed%2F&charset=%28detect+automatically%29&doctype=Inline&group=0 .

    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

    Comment by Michael P. Casey — June 2, 2013 @ 10:11 pm | Reply

    • 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.

      Comment by dmoskovich — July 9, 2013 @ 12:43 am | Reply

      • 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

        Comment by Michael P. Casey — July 9, 2013 @ 2:44 am

    • It passes now- thanks!!!

      Comment by dmoskovich — July 9, 2013 @ 4:35 am | Reply


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