Quantum invariants and low-dimensional topology: Hyperbolic volume and volume conjecture

Splice-unknotting and crosscap numbers.- Thomas Kindred (University of Nebraska at Lincoln)

Abstract: Ito-Takimura recently introduced the splice-unknotting number of a knot. This diagrammatic invariant provides an upper bound for a knot's crosscap number, with equality in the alternating case. Using results of Kalfagianni-Lee, this equality leads to corollaries regarding hyperbolic volume and the Jones polynomial.


Families of fundamental shadow links realized as links in $S^3$- Sanjay Kumar (Michigan State University)

Abstract: In 2015, Chen and Yang provided evidence that the asymptotics of the Turaev-Viro invariant of a hyperbolic $3$-manifold evaluated at the root of unity $\exp(\frac{2\pi i}{r})$ have growth rates given by the hyperbolic volume. This has been proven by Belletti, Detcherry, Kalfagianni, and Yang for an infinite family of hyperbolic links in connect sums of $S^1 \times S^2$ known as the fundamental shadow links. In this talk, I will present examples of links in $S^3$ satisfying the Turaev-Viro invariant volume conjecture through homeomorphisms with complements of fundamental shadow links along with an application towards the conjecture posed by Andersen, Masbaum, and Ueno (AMU conjecture).


Volume conjecture, geometric decomposition and deformation of hyperbolic structures (I) and (II)- Ka Ho Wong (Texas A & M University)



Abstract: The Chen-Yang volume conjecture of the Turaev-Viro invariant is a new topic in quantum topology. It has been shown that the $(2N+1)$-th Turaev-Viro invariant for a link complement can be expressed as a sum of norm squared of the colored Jones polynomial of the link evaluated at $t=\exp\left(\frac{2\pi i}{ N+\frac{1}{2}}\right)$. This leads to the study of the asymptotics for the $M$-th colored Jones polynomials of links evaluated at $(N+\frac{1}{2})$-th root of unity, with a fixed limiting ratio of $M$ to $N+\frac{1}{2}$. In the first talk, I will recall the definition of the colored Jones polynomials and discuss how the asymptotics of the colored Jones polynomials of the Whitehead link is related to the (not necessarily complete) hyperbolic structures on its complement. Then, in the second talk, we will focus on some satellite links whose complements have more than one hyperbolic piece in the geometric decomposition, and relate the asymptotics of their colored Jones polynomials to the geometric structures on the geometric pieces.

24 replies on “Quantum invariants and low-dimensional topology: Hyperbolic volume and volume conjecture”

Hi Ka Ho,

You have a very impressive list of results with respect to all different types of volume conjectures there! I will stop by your office hours, but I will post my questions here. In your first talk, I’m confused about the number $M$ as a sequence of $N$, how do you take the limit of a sequence over another sequence for the definition of $s$?

Also, could you say a little more about the holonomy of the meridian and the longitude and how you got the numbers $u$ and $v$ for the statement of the generalized volume conjectures for Whitehead chains?

Hi Christine,

Thank you for your questions. For the first one, we just take the limit in the usual sense, like what we do when we compute $\lim_{N\to \infty} \log N / e^N = 0$. For example, we may take $M$ to be the largest integer less than $0.9N$. Then the limiting ratio in this case will be $\lim_{N\to\infty} \lfloor 0.9N \rfloor / (N+1/2) =0.9$. You may replace $0.9$ by other number greater than $0$ to get a sequence $M$ with limiting ratio equal to that particular number.

For the second question about the choices of the holonomies, in the second talk (19:00-24:52), I discussed how to get the numbers u and v in the case of the colored Jones polynomials of the Whitehead link. The same argument works for the colored Jones polynomials of W_{0,1,c,0}. To be precise, we are able to find a triangulation of this link complement, together with a particular assignment of shape parameters, such that the critical point equations of the potential function corresponds to the hyperbolic gluing equations of the triangulation. Then, by using the differential equation satisfied by the potential function, we can see that the real part of the critical value is exactly the volume of the link complement equipped with that particular hyperbolic structure.

If you are interested in the physical interpretation of the generalized volume conjecture, you may also look at the paper `Three-Dimensial Quantum Gravity, Chern-Simons Theory, And The A-Polynomial’ by S. Gukov.

I hope this can answer your questions. I am also happy to talk with you in more details during the office hours 🙂

Hi Ka-Ho.

Nice talks! Sanjay Kumar whom you know has now found many families of links in the 3-sphere that satisfy that Turaev-Viro invariants volume conjecture (see his talk at this session and for more his recent preprint . Can we say something about (any version) of the Colored Jones polynomial VC for these links? Note, that Sanjay also showed that his links are in general distinct than Whitehead chains. Perhaps you too should talk about it?

Hi Effie,

Thank you for your comments!

For your question, I think the links also satisfy the generalized volume conjecture. First, the link complements discussed in Sanjay’s paper are homeomorphic to the fundamental shadow link complements. For the fundamental shadow links, in the paper ‘6j -symbols, hyperbolic structures and the volume
conjecture’ by F. Constantino, he showed that at the $N$-th root of unity $e^{\frac{2\pi i}{N}}$, the links satisfy the generalized volume conjecture (See Question 1.4 and Theorem 1.5 in So what we need to do is to prove similar result for the asymptotics of the 6j-symbols at the $(N+1/2)$-th root of unity.

In particular, we need to know the asymptotics of the quantum 6j symbols for general sequences of colors. I think you may be interested in the paper ‘Asymptotics of quantum 6j symbols’ by Q. Chen and J. Murakami (

You may need to fill in some details but it seems doable.

Constantino deals with the the “colored Jones polynomial”
of a link L in a 3-manifolD M that connect sums of S^2XS^1. This is, basically, the RT invariant of (M, L) where L colored in certain way.
Now suppose we have L_1 in S^3 so that S^3-L_1 is homeomorphic to M-L. In fact, often we get infinitely many links L_1 in S^3 for a given L. Now the TV invariant of S^3\L_1 is written as in terms of the multicolored CJP’s of L_1, but [DKY} as you said in your nice talks. These CJP’s are invariants of ambient isotopy of the links L_1 in S^3. It may be that the individual values can be different for different links L_1 (even though the links have the same complement). Why would these CJP’s would be the same as the invariant considered by Constantino? Do you have a way of seeing this?

I think you’re right. I just messed up (and everybody can see that I messed up 😛 )

I don’t have any idea about how to do it now. Nevertheless, the question about the VC of the CJP of these links sounds interesting. Is it possible to find formulas for the CJP’s of these links in terms of the 6j-symbols?

Hi Effie, Sanjay and Ka Ho,
I believe Effie may be asking about something I did in my PhD thesis.
Volume conjecture for knotted trivalent graphs (2008)
The knotted trivalent graphs introduced there include a large class of links (not graphs) in S^3 whose complement coincides with the complements of the fundamental shadow links. The links I describe also include the links Sanjay constructs in his recent paper you mention. For example his thm 4.1 is very close to my corollary 1.
(As a minor technical detail I added some additional rings to make the analysis trivial) but I think there might be still be some useful stuff in my paper for people working on colored Jones and volume conjecture. The knotted trivalent graphs may seem foreign at first but they are very helpful and carry beautiful geometric structures. They include knots and links but provide additional nice operations that one does not have for links. See also Dylan Thurston’s excellent The algebra of knotted trivalent graphs and Turaev’s shadow world

Hi Rolland, I’m inclined to think that. your paper discusses links that are close (if not the same) to the links considered by Belletti in Getting the doubles of the neighborhoods of graphs you consider gives FSL (see belletti).
The links though live in copies of S^2XS^1

Sanjay’s Thm 4.1 is related to bakers construction and the augmented links should be related to yours as well. I don’t know that the explicit families in Sanjay constructs in S^3 are appearing in your paper. For those we still need to calculate the cJP to prove the original VC… Actually do you know, for which links in S^3 does your construction apply to–I’m talking without the augmentation…

No Efffie, the links I described are all in the three sphere. You get links from KTGs by unzipping edges until no more vertices are left. If you add a ring every time you untwist you get in particular Baker’s pictures.
As a basic example try constructing the Borromean rings this way, starting from a tetrahedron graph, it’s fun! In summary I believe the KTG approach is a nice and clean set up for discussing links in the 3-sphere covering all the cases that Sanjay was talking about.

Hi Roland,

thanks for your message. Yes, I understand your links are in S^3, and after augmenting you can create hyperbolic octahedral links.
The question is which of these links have complements that are homeomorphic to FSL, for which we know the asymptotics of the TV invariants. Sanjay works with shadows to identify links in S^3 with complements isomorphic to FSL– the constructions are explicit enough that also give information about the topology of the links (e.g. fibrations) and new info about the AMU conjecture. I don’t expect that all the FSL appear in S^3 (there are knots among the FSL and don’t know of knots with volume 2v_8 in S^3). With Sanjay we’d like to understand which links in S^3 have complements that are ism to these of FSL. I think it would be nice to know which of the links constructed from KTG’s also have this property. For example, the augmented 2-bridge links (=all octahedral fully augmented links) fit into the KTG scheme you consider. Do you know if they have FSL complements?

Hi Effie, Roland and Sanjay,

I have a question about the links constructed in Sanjay’s paper. I think the families L_k, J_k and K_k are included in the links considered in Roland’s paper, but I am not sure whether the family b_k is also included. I tried to draw the diagram of b_1 but I don’t know how to get this from applying unzipping move on some trivalent graph. Could you say more about this? Thanks!

Hello Roland,

Thank you for your comments and the references. I had a question in regards to the KTGs. Please correct me if I am wrong, but if I understand correctly, all of the KTGs can be obtained from performing the KTG moves on a standard tetrahedron. The volume of the singly augmented KTG will be 2(t+2)v_8 where t is the number of triangle moves. If we want to consider single augmented KTGs with volume 2v_8, do we only consider augmentations on the standard tetrahedron? Also in this case, do we obtain all six FSL in S^3 of volume 2v_8 that I mention in my talk (Theorem 3.4 of paper)?

Hello everyone, sorry for the late introduction. My name is Sanjay Kumar, and I am a graduate student at Michigan State University with Effie Kalfagianni as my advisor. If you have any questions on my talk, please feel free to leave a comment. Although I do not have any official office hours, I am happy to speak via Zoom. Just leave a comment or send me an email, and we can schedule a meeting.

Hi Ka-Ho (Sanjay and Rolland),

It is no surprise that the first family of Sunjay’s also fit in Rolands scheme: Topologically are obtained by ”belt summing” complements of links in the Borromean family. This goes back to our discussions in Da Nhang Vietnam last May. The point is that the only way we know to prove the TV invariants volume conjecture for these links is to identify their complements as these of FSL. I don’t think we know how to prove Kashaev’s conjecture either if w don’t augment further by rings as Roland did. The interesting direction, in my mind, is which octahedral links in S^3 have complements homeomorphic to these of FSL (which basically allows to calculate the TV invariant)

Hi Sanjay,
Yes I think you’re right and it looks like there are fundamental shadow link complements that are not covered by the 1-augmented KTGs I considered in my paper. I would still appreciate it if you add a reference to my KTG paper in your latest work.

Hello Roland,

Thank you once again, I will make sure to update my paper to include your work with the KTG.

Rolland, yes this was an omission and I’m to blame–I should have noted:-( –thanks for your comments that made me also think and clarify some things for my self…

Hi Effie,

I think I know how to prove the VC of the CJP of the hyperbolic links obtained from Roland’s construction at $t=e^{\frac{2\pi i}{N+1/2}}$. At this root, the trick in Roland’s paper (Lemma 2) does not work. I need some time to write down the details. In short, the strategy of proving the VC of CJP of Whitehead chains also work for the hyperbolic links discussed in Roland’s paper. As a corollary, this also implies the VC for the Turaev-Viro invariants of the link complements. This gives an alternative proof of Sanjay’s results about the VC of L_k, J_k and K_k (Theorem 3.1). I am not sure whether it can imply the result for b_k due to the question I mentioned before.

Besides, for the generalized VC, I am studying the triangulation of the link complement, which is also discussed in Roland’s paper. We may also need to know more about the geometry of the 6j symbol. I will let you know when I finish the draft.

Ηι Κα-Ηο,

1. I don’t know that proving. the CJP for t as you say, proves the Tv volume conjecture…you need to control the whole sum in DKY

2. Also we don’t know what you mean ~the links in Rolands paper~ : I don’t have a characterization for that…A concrete question would be to prove the Vc for the octahedral fully augmented links. This is the question I asked you and Sanjay to work on in Vietnam. the question is still open.

Finally , rather than discussing here you. can. write. your thoughts and send them to us.

Yep. It is better to discuss through email (or shall we have a zoom meeting some time this week?). The following is a short reply.

1. I am sorry that I didn’t explain it clearly. To be precise, I mean the strategy described in Section 2.3 of my second preprint ( works.

2. The links in Roland’s paper is obtained as follows: given a KTG, keep applying the 1-unzip on it until there is no vertices. I think the resulting link is an octahedral fully augmented link. But I am not sure whether all octahedral fully augmented link can be obtained in this way.

Since this part of the conference is on the volume conjecture I’d like to share a thought on this topic. Let me start with two points:
1) Did you notice that many of the conjectures about colored Jones are about some kind of limiting behaviour as N grows?
(generalized) Volume conjecture but also AJ and slope conjecture are like this.
2) Also the best way I know to define and work with the colored Jones polynomial is to apply the irreducible N-dimensional representation to the universal quantum sl_2 invariant of Lawrence, Ohtsuki and Habiro. (For example Habiro found all his groundbreaking work on colored Jones this way!)
Putting together 1) and 2) I would like to speculate (as one does at conferences) whether it would be possible to ditch the N-dimensional representations and reformulate the volume conjecture and other colored Jones conjectures directly in terms of the universal quantum sl_2 invariant.
This may seem even more complicated and forbidding, however quantum sl_2 is quite close (dual!) to the (deformed) algebra of functions on the group SL(2,C). As such it can be used to discuss hyperbolic geometry and other classical SL(2,C) notions directly.
Of course what I am saying here is not new and some aspects were explored in many works including Baseilhac-Benedetti, Frohman, Le, Rozansky and others. But since my efforts on volume conjecture often got lost in special families and hard analysis (or both) I would like to point out once more that perhaps the representations are holding us back and the Kauffman bracket skein relation may not be your best friend after all.

Hi Thomas, Sanjay and Ka Ho!
I enjoyed watching your talks, and learned lots of things from them!

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