Last week, Stavros Garoufalidis posted [1] on ArXiv, which got me daydreaming:

The coloured Jones function is an invariant of a knot consisting of a sequence of elements in indexed by . The boundary slope of an incompressible surface is where and are the number of times the surface winds around the meridian and longitude of correspondingly. The conjecture is that the limit points of are boundary slopes of incompressible surfaces in . One doesn’t need to know much quantum topology to find this intriguing… why should boundary slopes of incompressible surfaces be visible in such a straightforward way to the coloured Jones function?

The -character variety is the only common ground I can see. I will summarize it as follows:

A decomposition of the fundamental group of a 3-manifold as a free product with amalgamation or an HNN decomposition corresponds (non-canonically) to a system of incompressible surfaces in . If is hyperbolic, you can get such a decomposition by embedding in and piggy-backing on deep results of Tits-Bass-Serre on the structure of subgroups of . This is a deep idea in itself whose primitive form is due to Stallings, that was used plenty pre-Culler-Shalen. The problem is that you know nothing about the incompressible surfaces- in particular they might be boundary parallel.

Enter Culler-Shalen [2], who had the idea of looking at all the representations of in at the same time (actually their characters) which form a variety called the -character variety of . Ideal points of a smooth projective model of a curve in give rise to non-boundary-parallel incompressible surfaces in .

I don’t know how the -character variety of M sees slopes (do you, reader?). But surely that must be a part of the story. The -character variety and the Jones polynomial are related through the -polynomial.

## December 2, 2009

### Garoufalidis’s Jones Slope conjecture

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## 23 Comments »

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Hi Dan – your coblogger knows all about how the SL_2(C) character variety sees slopes! Geometrically the story is as follows: the complete hyperbolic structure on a cusped 3-manifold can be deformed to incomplete structures. As one deforms infinitely far, the space of hyperbolic structures can be compactified by an action of your fundamental group on an R-tree; for suitable degenerations, this R-tree is a genuine tree and therefore dual to some essential surface (this is the “slope” associated to the degeneration). Algebraically, one thinks of a representation of pi_1 into SL(2,A) where A is some subring of the field of meromorphic functions on a curve in the character variety. An “ideal point” on this curve determines a discrete valuation on A, and therefore an action of SL(2,A) on the associated Bass-Serre tree (dual to a slope etc.) I think it is known that there are examples of *undetected* slopes (there is an old paper of Cooper-Long claiming this, but I am pretty sure it is known to be wrong; nevertheless, I think I remember that there are other (more correct) examples). As you say, the A polynomial encodes this information, more or less. Incidentally, the “slopes” that are detected by ideal points can be read off from the Newton polygon of the curve – they correspond to the “slopes” (i.e. the sides) of the Newton polygon! (this is in CCGLS – again, ask your coblogger)

Best,

Danny

Comment by Danny Calegari — December 2, 2009 @ 11:48 am |

Thanks!! Clearly I must read CCGLS urgently (for many reasons)…

Comment by Daniel Moskovich — December 6, 2009 @ 10:59 pm |

What is the motivation for studying the coloured Jones polynomial? Whenever I hear reference to it, there’s the associated words “Volume conjecture” or this “slope conjecture”. Is it generally attributed to Turaev, as in the Garoufalidis paper?

Comment by Ryan Budney — December 3, 2009 @ 2:53 am |

While I’m asking the obvious questions, one more. There’s been quite a bit of effort towards trying to find a topological or geometric interpretation of aspects of the Jones polynomial.

To what extent has the question been turned on its head? Is there a strong “physical” interpretation of the Alexander polynomial? Exactly how concretely physical can you make it — is there an actual physical system that you could construct in a lab where the Alexander polynomial of a knot would be relevant to the dynamics of the system?

Comment by rybu — December 3, 2009 @ 3:21 am |

Let me answer in tiny pieces… the Melvin-Morton-Rozansky conjecture, proved by Bar-Natan and Garoufalidis, states that if you expand the colored Jones polynomial of a knot in powers of (q-1), then the bottom line in this expansion is equal to the inverse Alexander polynomial of the knot. This might be thought of as a strong “physical” interpretation in terms of Chern-Simons path integrals over SU(2) connections in the knot complement…

I think the coloured Jones is interesting not because of the volume conjecture, but because it suggests a new algebraic topology for 3 dimensions. Homology, homotopy, Reidemeister torsion etc. were

inventedto solve problems in arbitrary dimensions, while the coloured Jones wasdiscovered. Its unreasonable effectiveness distinguishing knots, and its (conjectured) relationship with hyperbolic geometry in 3-dimensions suggests that we might be in the process of uncovering a whole new way of analyzing objects in dimension 3, which may be better suited to dimension 3 than plain old homology and homotopy.I don’t think this dream is anywhere near being realized yet… but I think this is the secret dream of every quantum topologist.

The Alexander polynomial might be a case-in-point… it turns up (as if by magic) as the most basic object in the study of the coloured Jones (in some sense), suggesting that the coloured Jones might be its natural extension, which sees much deeper into the topology than just the bounding lines in its expansion. Also, the strong relationship with the character variety.

So, as Atiyah put it, the coloured Jones was not designed, but discovered, and seems to have many wonderful properties. The goal, rather than to use it to prove other stuff, is to understand where it comes from topologically and to explain these marvelous properties, in the hope of developing a better understanding of topology in dimension 3, with all its special features.

Comment by Daniel Moskovich — December 3, 2009 @ 5:19 am |

Hi Daniel,

Melvin-Morton-Rozanzky, is that where coloured Jones originated? I’m asking where the first occurance of the definition is in the literature and why it appeared.

Comment by rybu — December 3, 2009 @ 9:41 am

I’d say that colored-Jones provides a great example of a quantum invariant, but if you’ve not already interested in quantum topology I suppose that’s not much motivation. IIRC, the volume conjecture is a connection between the coefficients of colored-Jones and the hyperbolic volume of the complement of a hyperbolic knot. I’m not sure about a “physical” interpretation of the Alexander polynomial, but it’s got a very clear meaning in terms of first-year algebraic topology, which is why

certainclassical knot theorists accept it while disdaining quantum invariants that are defined essentially by means of skein relations.Basically, you take the complement of the knot. The knot group is the fundamental group of this space. Given an orientation of the knot, the linking number gives a homomorphism , which of course has a kernel which is a normal subgroup of . Corresponding to this subgroup we have a covering space, which we call the infinite cyclic cover. The integers then act by deck transformations on this covering space, which makes its homology into a module over the group algebra of the integers — Laurent polynomials. The module has an annihilator, which is principal, and any generator of this ideal is called the Alexander polynomial, which is thus well-defined up to sign and power of its variable.

Comment by John Armstrong — December 3, 2009 @ 4:10 am |

I would argue that quantum invariants are defined essentially by representation theory of quantum algebra (ribbon Hopf algebras, and quantum groups in particular) as opposed to by skein relations. For instance, it was a long time before skein relations were known for the Links-Gould invariant. Does every quantum invariant satisfy a skein relation?

Comment by Daniel Moskovich — December 3, 2009 @ 5:29 am |

Hi John,

I’m okay with Alexander’s definition of the Alexander polynomial. What I’m asking is how far you can go in the other direction — to what extent is the Alexander polynomial a “quantum invariant”? Saying it satisfies skein relations IMO is very far from saying it has anything to do with physics. Something that satisfies skein relations I suppose I would say is a combinatorial object defined via Reidemeister moves.

Back to my question: Is there an explicit physical system you could set up in a laboratory, where you would craft a knot out of some material, take some measurement with some physical apparatus and measure aspects of the Alexander polynomial of that knot? ie can a physical system “compute” the Alexander polynomial in a natural way?

Comment by rybu — December 3, 2009 @ 9:49 am |

Daniel, you’re right that skein relations are not the is-all, end-all of quantum topology, but I think we’re using the terms slightly differently. I’m meaning a more general sense in which we talk about invariants arising from the algebraic structure of tangles as a braided monoidal category (like quantum group invariants do). And rybu is correct that these skein relations are further from physics and classical topology, and closer to combinatorial objects.

But that’s also sort of my point in that one aside: I’ve run into very well-known classical knot theorists who have literally said in so many words that they don’t regard the Jones polynomial and the whole field that followed from it as topology. Alexander may have a definition from skein relations, and this helps us compute it, but it’s “really” this topological construction that “has something to do with” the knot and how it sits in three-dimensional space. Jones comes from combinatorics on knot diagrams, and I don’t think anyone has a good idea of what it has to do with how a physical string might sit in space. If anything, quantum invariants are further from physics than classical ones.

I think, rybu, you’re mistaking “quantum” in “quantum topology” as having anything to do with physics. As my advisor was fond of saying, “quantum groups are neither quantum nor groups”. It’s really more to do with the fact that they work with a non-Cartesian monoidal product on the relevant category, which ultimately has some connection to quantum physics.

Comment by John Armstrong — December 3, 2009 @ 12:42 pm

I think MMR answers that. You build a knot complement in a box, set up an SU(2) Chern-Simons action in the box, and you can observe the Alexander polynomial.

There are other answers too. First, the Alexander polynomial is (part of) the universal finite type invariant for the Lie superalgebra (Figueroa-O’Farill, Kimura, Vaintrob), so if you could make your supersymmetry work like , you would have the Alexander polynomial right there. Vaintrob has a paper where he explains that this actually implies MMR.

One more option, which I don’t understand at all, is Rozansky and Saleur’s interpretation of the Alexander polynomial in terms of the WZW model. I presume this would be the “physicist’s choice”, because it is the one I understand least about. In that context, again you could observe the Alexander polynomial in a theoretical lab.

Comment by Daniel Moskovich — December 6, 2009 @ 10:53 pm

I should say more… I would argue that the Alexander polynomial is not only a quantum invariant, but the prototypical quantum invariant. This is the main reason I care about the Alexander polynomial (!)… I don’t completely agree with what John said- assuming vanishing anomalies, path integrals, and other physicist magic, saying something is the universal finite-type invariant for some Lie superalgebra is the same as saying it’s a partition function, which encodes the statistical mechanics of the space. So the Alexander polynomial would give you pressure, entropy, free energy, and other important statistical mechanical quantities… of a manifold equipped with G-gauge fields where G is the simplest Lie superalgebra you can think of ( is my vote).

Comment by Daniel Moskovich — December 7, 2009 @ 3:46 am

Hi Daniel et al-

I like character varieties as much as the next hyperbolic geometer, but let me suggest a more low-tech interpretation of Stavros’s conjecture. Associated to a knot diagram with crossings, there are Kauffman states — ways to smooth the diagram near each crossing. The Jones polynomial can be expressed as a sum of monomials corresponding to these states, so there is some state that gives the highest power of .

But each state of also corresponds to a surface . The state consists of a bunch of circles, so have each circle bound a disk, and connect by half-twisted bands. It’s exactly the same as Seifert’s construction of a Seifert surface, except our construction might give something non-orientable. The slope of the state surface can be easily read off — essentially it’s the (signed) number of crossings at which deviates from the Seifert state. It’s also the power of the corresponding monomial in .

The punchline is that these state surfaces are frequently incompressible. See, for example, Ozawa (arXiv:0609.5166). Under “nice” hypotheses, the surface corresponding to the highest power of will be essential. The other punchline is that (as Stavros observes in his paper) the top power of the colored Jones polynomial is quite well-behaved as we take more and more cables, and . So what we see in the diagram for the original polynomial might reasonably predict the asymptotic picture.

Now: the previous paragraph involved all sorts of wishful thinking. The “nice” hypotheses don’t always hold, and the argument doesn’t always work. But then, sketched arguments using the character variety also involve a certain amount of magical thinking, because of undetected slopes, etc.

Comment by Dave Futer — December 4, 2009 @ 9:05 am |

Thanks Dave… this is beautiful and simple.

Comment by Daniel Moskovich — December 6, 2009 @ 10:57 pm |

Thanks, Daniel! I should add that the approach I’m describing above is actually now carried out for adequate knots. See arXiv:1002.0256.

Comment by Dave Futer — February 3, 2010 @ 1:26 pm

Is “The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C)” by Kirby and Melvin where the original definition of the coloured Jones polynomial appears?

Comment by Ryan Budney — December 4, 2009 @ 9:06 pm |

I think the answer is no… the coloured Jones polynomial is just a specialization of the HOMFLYPT, so I haven’t heard anyone claim credit… one early source is Murakami, Jun “The parallel version of polynomial invariants of links.”

Osaka J. Math. 26 (1989), no. 1, 1–55. which discusses the coloured Jones polynomial. The motivation given there is that looking at a knot and all its -cables gives more information than looking just at the link (from the point of view of the Jones polynomial), and that cabling is a natural operation from the point of view of the quantum algebra definition (it’s the diagonal map- comultiplication) and “stabilizes”. That’s what you’re seeing with all these conjectures relating “infinite cables” with hyperbolic information… that if you stabilize enough times, the coloured Jones polynomial becomes a combination of geometric quantities.

You could trace this insight back to the beginning of quantum topology- Kontsevich, Lin, and Bar-Natan (and probably quite a few others) noticed that cabling “stabilizes” links in the quantum sense.

Comment by Daniel Moskovich — December 6, 2009 @ 10:37 pm |

Another ignorant question — is the Alexander module a quantum invariant? Are things like the Milnor signature invariants and the Tristram-Levigne invariants extractable from something like the coloured Jones polynomial?

Comment by Ryan Budney — December 14, 2009 @ 4:46 pm |

This is a very nice question, which I should write a proper post to answer… let’s distinguish between three things:

Quantum invariants: Associated to an R-matrix, coming from a representation of a quantum group (or a ribbon Hopf algebra). The (coloured) Jones polynomial and the Alexander polynomial are both examples of quantum invariants, associated to different quantum groups.Finite type invariants: Associated to Jacobi diagrams (formal finite sums of graphs) plus a “weight system”, which in this context is a map from Jacobi diagrams to satisfying certain relations. Coefficients of the coloured Jones polynomial is an example, or coefficients of the Alexander polynomial.The Kontsevich invariant: A formal sum of Jacobi diagrams over . It is universal with respect to both finite type invariants and quantum invariants. A weight system gives rules to contract Jacobi diagrams and get numbers, and you can get any finite type invariant this way. An R-matrix gives (closely related) rules to contract Jacobi diagrams and get a polynomial (for example), and you can get any quantum invariant this way.This is what I think John meant- the Alexander polynomial is a quantum invariant, but the Blanchfield pairing is not. However, the Blanchfield pairing

isa (very important) part of the Kontsevich invariant. This is why MMR is true conceptually, and once you see this, MMR becomes easy.The Blanchfield pairing corresponds to the coefficients of Jacobi diagrams with precisely one loop (the “wheels part” of the Kontsevich invariant). If you choose an R-matrix, these will contract and you will be left with the Alexander polynomial (or some reduction of it)- the presentation matrix will collapse to its (Kadison-Fuglede) determinant. That’s how you prove MMR.

The signature function plays an important conjectural part in the theory of the coloured Jones polynomial. The key paper is http://arxiv.org/abs/math/0310203. Additionally, the Casson invariant of the infinite cyclic cover can be recovered from the expansion of the coloured Jones of a knot in powers of (q-1) as the “second line” (the bottom line is the Alexander polynomial). The Casson invariant of the infinite cyclic cover contains the “total signature”.

Stoimenow seems to have written a lot about Tristram-Levine signatures and finite type invariants, which I do not understand well conceptually. I assume an answer to the conjecture hinted at in the previous paragraph would make that all clear.

These are some of the questions I am most interested in in quantum topology, and a good mathematical understanding of the role of signatures, Minkowski units, Nakanishi index, and such would really clarify the coloured Jones polynomial as it relates to classical topology (I think).

Comment by Daniel Moskovich — December 14, 2009 @ 8:54 pm |

And another!

The result (apparently due to Bar Natan) that the Alexander polynomial is determined by the coloured Jones polynomial, can it been souped-up to give a proof of the symmetry of the Alexander polynomial? ie: can you “lift” Poincare duality to a theorem at the level of the coloured Jones polynomial?

Comment by Ryan Budney — December 14, 2009 @ 4:53 pm |

Another super question!

I have no idea. The Alexander polynomial is a Reidemeister torsion in a natural way (and so is the hyperbolic volume), but no K-theory interpretation of the coloured Jones polynomial is known, and I assume that you would need something of this sort to “lift” Poincare duality.

I wonder what the physicists say…

Comment by dmoskovich — December 14, 2009 @ 9:03 pm |

I’m not sure whether Dan would call the Alexander module a quantum invariant, but I can tell you that it

isrelated to the representation theory of the category of tangles. But not in the same way that Jones/bracket is.Comment by John Armstrong — December 14, 2009 @ 4:57 pm |

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Avril Smith

Comment by cim gfe — May 16, 2010 @ 8:27 pm |