Volume conjecture: refined and categorified. (English) Zbl 1282.57016

Let \(J_n(K;q)\) be the colored Jones polynomial of a knot \(K\) in the three-sphere associated with the \(n\)-dimensional irreducible representation of the Lie algebra \(sl(2;\mathbb{C})\). The volume conjecture ([R. M. Kashaev, Lett. Math. Phys. 39, No. 3, 269–275 (1997; Zbl 0876.57007); H. Murakami and J. Murakami, Acta Math. 186, No. 1, 85–104 (2001; Zbl 0983.57009)]) states that the limit \(\lim_{n\to\infty}\frac{2\pi\log|J_n(K;e^{2\pi\sqrt{-1}/n})|}{n}\) would coincide with the volume of the knot complement. Given a complex parameter \(u\), this can be generalized to the generalized volume conjecture \(J_n(K;e^{\hbar})\underset{{n\to\infty}\atop{\hbar\to0}}{\sim}\exp\left(\frac{1}{\hbar}S_0(u)+\sum_{k\geq0}S_{k+1}(u)\hbar^k\right)\), where \(\hbar\) satisfies \(e^{n\hbar}=e^{u}\) and \(S_0(u)\) is expected to define the Chern–Simons invariant of a representation of the fundamental group of the knot complement to \(SL(2;\mathbb{C})\) parameterized by \(u\). ([S. Gukov, Commun. Math. Phys. 255, No. 3, 577–627 (2005; Zbl 1115.57009); H. Murakami, Adv. Math. 211, No. 2, 678–683 (2007; Zbl 1121.57006); S. Gukov and H. Murakami, Fields Institute Communications 54, 261–277 (2008; Zbl 1171.57012)]). When \(u=0\), it is a restatement of the (complexified) volume conjecture [H. Murakami et al., Exp. Math. 11, No. 3, 427–435 (2002; Zbl 1117.57300)]. The parameter \(u\) is regarded as a deformation parameter in the case of a hyperbolic knot. The eigenvalues of the image of the meridian by the associated representation are \(e^{\pm u}\). If we denote the eigenvalues of the image of the longitude by \(e^{\pm v}\), then \(v\) coincides with \(d\,S_{0}(u)/d\,u\) (see [W. D. Neumann and D. Zagier, Topology 24, 307–332 (1985; Zbl 0589.57015)]). Therefore \(S_{0}(u)\) is expected to be given as the integral \(\int v\,du=\int\log(y)\frac{dx}{x}\), where \((x,y):=(e^u,e^v)\) is in the zero locus of the \(A\)-polynomial \(A_K(x,y)\) [D. Cooper et al., Invent. Math. 118, No. 1, 47–84 (1994; Zbl 0842.57013)].
It was observed in [S. Gukov, loc. cit.] that the generalized volume conjecture is closely related to the AJ-conjecture, S. Garoufalidis [Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 291–309 (2004; Zbl 1080.57014)], which states that the colored Jones polynomial is annihilated by a quantum version of the \(A\)-polynomial \(\hat{A}_K(\hat{x},\hat{y};q)\). Precisely speaking, for any knot \(K\) there exists a non-commutative polynomial \(\hat{A}_K(\hat{x},\hat{y};q)\) such that \(\hat{A}_K(\hat{x},\hat{y};q)J_n(K;q)=0\) and that \(A_K(x,y):=\hat{A}_K(\hat{x},\hat{y};q)\big|_{q=1,\hat{x}=x,\hat{y}=y}\) coincides with the \(A\)-polynomial, where \(\hat{x}\) acts on \(J_n(K;q)\) by multiplication by \(q^n\) and \(\hat{y}\) replaces \(n\) with \(n+1\). Note the commuting relation \(\hat{y}\hat{x}=q\hat{x}\hat{y}\). This conjecture is called the quantum volume conjecture in this paper.
In the paper under review the authors propose further generalizations of the above conjectures. Let \(P_{n}(K;q,t)\) be the Poincaré polynomial of a categorification \(\mathcal{H}^{n}_{i,j}(K)\) of the colored Jones polynomial \(J_n(K;q)\), that is, \(P_{n}(K;q,t)\) is defined as \(\sum_{i,j}q^it^j\dim\mathcal{H}^{n}_{i,j}(K)\) and \(P_{n}(K;q,-1)\) coincides with \(J_n(K;q)\). Then the refined (or categorified) quantum volume conjecture states that there would exist a non-commutative polynomial \(\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,t)\) such that \(\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,t)P_{n}(K;q,t)\) vanishes and that \(\hat{A}_K(\hat{x},\hat{y};q)=\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,-1)\). The refined generalized volume conjecture states that the asymptotic expansion of \(P_n(K;e^{\hbar},t)\) with respect to \(\hbar\) would become \(P_n(K;e^{\hbar},t)\underset{n\to\infty}{\sim}\exp\left(\frac{1}{\hbar}S_0(u,t)+\sum_{k\geq0}S_{k+1}(u,t)\hbar^k\right)\) such that \(S_0(u,t)\) is given by the integral \(\int\log(y)\frac{dx}{x}\), where \((x,y)\) is in the zero locus of a deformation (or a categorification) \(A^{\roman{ref}}_K(x,y;t):=A^{\roman{ref}}_K(x,y;1,t)\) of the \(A\)-polynomial.
The authors calculate \(\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,t)\) and \(A^{\roman{ref}}_K(x,y;t)\) for the unknot and the torus knots of type \((2,2p+1)\) by using various techniques from physics.


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
81T13 Yang-Mills and other gauge theories in quantum field theory
58J28 Eta-invariants, Chern-Simons invariants
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