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 . 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 be a long knot and let be its topological moduli space. There are two natural loops in : the rotation of around the long axis, which we call and another loop, which we call Hatchers loop (compare ). It is defined as follows: one puts a pearl (i.e. a small 3-ball B) on the (closure of the) knot in the 3-sphere. The part of in is a long knot. Pushing B once along the knot induces Hatcher’s loop in . The following theorem is an immediate consequence of a result of Hatcher (see ).
Theorem 1 (Hatcher)
Let be a long knot which is not a satellite (i.e. there is no incompressible torus in its complement in the 3-sphere). Then and represent the same class in if and only if 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  and  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 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?
R(1) seems genuinely new, deep, and worthy of substantial further study. This is tremendously exciting!