We use Dehn surgery methods to construct infinite families of hyperbolic knots in the 3-sphere satisfying a weak form of the Turaev–Viro invariants volume conjecture. The results have applications to a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. We obtain an explicit family of pseudo-Anosov mapping classes acting on surfaces of any genus and with one boundary component that satisfy the conjecture.

Using Ohtsuki’s method, we prove the asymptotic expansion conjecture and the volume conjecture of the Reshetikhin–Turaev and the Turaev–Viro invariants for all hyperbolic 3-manifolds obtained by doing a Dehn-surgery along the figure-8 knot.

We show the n n colored Jones polynomials of a highly twisted link approach the Kauffman bracket of an n n colored skein element. This is in the sense that the corresponding categorifications of the colored Jones polynomials approach the categorification of the Kauffman bracket of the skein element in a direct limit, as the number of full twists of each twist region tends toward infinity, proving a quantum version of Thurston’s hyperbolic Dehn surgery theorem implicit in Rozansky’s work, and giving a categorical version of a result by Champanerkar-Kofman [Algebr. Geom. Topol. 5 (2005), pp. 1–22]. In view of the volume conjecture, we compute the asymptotic growth rate of the Kauffman bracket of the limiting skein element at a root of unity and relate it to the volumes of regular ideal octahedra that arise naturally from the evaluation of the colored Jones polynomials of the link.

This paper proves quantum modularity of both functions from $\mathbb{Q}$ and $q$-series associated to the closed manifold obtained by $-\frac{1}{2}$ surgery on the figure-eight knot, $4_1(-1,2)$. In a sense, this is a companion to work of Garoufalidis-Zagier, where similar statements were studied in detail for some simple knots. It is shown that quantum modularity for closed manifolds provides a unification of Chen-Yang's volume conjecture with Witten's asymptotic expansion conjecture. Additionally we show that $4_1(-1,2)$ is a counterexample to previous conjectures of Gukov-Manolescu relating the Witten-Reshetikhin-Turaev invariant and the $\widehat{Z}(q)$ series. This could be reformulated in terms of a ''strange identity'', which gives a volume conjecture for the $\widehat{Z}$ invariant. Using factorisation of state integrals, we give conjectural but precise $q$-hypergeometric formulae for generating series of Stokes constants of this manifold. We find that the generating series of Stokes constants is related to the 3d index of $4_1(-1,2)$ proposed by Gang-Yonekura. This extends the equivalent conjecture of Garoufalidis-Gu-Marino for knots to closed manifolds. This work appeared in a similar form in the author's Ph.D. Thesis.

Abstract We automate the process of machine learning correlations between knot invariants. For nearly 200 000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural network on the input invariants. Correlation between invariants is measured by the accuracy of the neural network prediction, and bipartite or tripartite correlations are sequentially filtered from the input invariant sets so that experiments with larger input sets are checking for true multipartite correlation. We rediscover several known relationships between polynomial, homological, and hyperbolic knot invariants, while also finding novel correlations which are not explained by known results in knot theory. These unexplained correlations strengthen previous observations concerning links between Khovanov and knot Floer homology. Our results also point to a new connection between quantum algebraic and hyperbolic invariants, similar to the generalized volume conjecture.

Matrix-valued holomorphic quantum modular forms are intricate objects associated to 3-manifolds (in particular to knot complements) that arise in successive refinements of the volume conjecture of knots and involve three holomorphic, asymptotic and arithmetic realizations. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the (-2,3,7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-2,3,7)$$\\end{document}-pretzel knot.

We construct a new infinite family of ideal triangulations and H–triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichmüller TQFT and to prove the associated volume conjecture for all twist knots, using the saddle point method.

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston’s gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a boundary-parabolic [Formula: see text]-representation to boundary-unipotent [Formula: see text]-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.

In this paper, we study the variation of the Turaev–Viro invariants for [Formula: see text]-manifolds with toroidal boundary under the operation of attaching a [Formula: see text]-cable space. We apply our results to a conjecture of Chen and Yang which relates the asymptotics of the Turaev–Viro invariants to the simplicial volume of a compact oriented [Formula: see text]-manifold. For [Formula: see text] and [Formula: see text] coprime, we show that the Chen–Yang volume conjecture is stable under [Formula: see text]-cabling. We achieve our results by studying the linear operator [Formula: see text] associated to the torus knot cable spaces by the Reshetikhin–Turaev [Formula: see text]-Topological Quantum Field Theory (TQFT), where the TQFT is well-known to be closely related to the desired Turaev–Viro invariants. In particular, our utilized method relies on the invertibility of the linear operator for which we provide necessary and sufficient conditions.

The volume conjecture for polyhedra implies the Stoker conjecture

We define a polynomial invariant $J_n^T$ of links in the thickened torus. We call $J^T_n$ the $n$th toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, $J_n^T$ exhibits volume conjecture behavior. We prove the volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions of $J_n^T$, one as a generalized operator invariant we call a pseudo-operator invariant, and another using the Kauffman bracket skein module of the torus. Finally, we show $J^T_n$ produces invariants of biperiodic and virtual links. To our knowledge, $J^T_n$ gives the first example of volume conjecture behavior in a virtual (non-classical) link.

Abstract We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3‐manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3‐manifolds with totally geodesic boundary.

Based on the complexity equals action (CA) and complexity equals volume (CV) conjectures, we investigate the holographic complexity of a slowly accelerating Kerr-AdS black hole in the bulk Einstein gravity theory which is dual to holographic states with rotation and conical deficits in the boundary quantum system. Upon obtaining an implicit form of the Wheeler-DeWitt patch, we evaluate the action and show that the growth rate of the CA complexity violates volume-scaling formulation in large black hole limit due to the non-trivial contribution from the not-too-small acceleration of the black hole. Moreover, in an ensemble with fixed entropy, pressure, and angular momentum, we also find that complexity of formation decreases with both the average and difference of the conical deficits on the poles when the black hole is close to the static limit but increases with the deficits when the black hole is close to the extremal regime.

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

Volume Conjecture and WKB Asymptotics

In this paper, we show that the Turaev-Viro invariant volume conjecture posed by Chen and Yang is preserved under gluings of toroidal boundary components for a family of $3$-manifolds. In particular, we show that the asymptotics of the Turaev-Viro invariants are additive under certain gluings of elementary pieces arising from a construction of hyperbolic cusped $3$-manifolds due to Agol. The gluings of the elementary pieces are known to be additive with respect to the simplicial volume. This allows us to construct families of manifolds with an arbitrary number of hyperbolic pieces such that the resultant manifolds satisfy an extended version of the Turaev-Viro invariant volume conjecture.

We study a variant of quantum circuit complexity, the binding complexity: Consider an $n$-qubit system divided into two sets of ${k}_{1}, {k}_{2}$ qubits each (${k}_{1}\ensuremath{\le}{k}_{2}$) and gates within each set are free; what is the least cost of two-qubit gates ``straddling'' the sets for preparing an arbitrary quantum state, assuming no ancilla qubits allowed? First, our work suggests that, without making assumptions on the entanglement spectrum, $\mathrm{\ensuremath{\Theta}}({2}^{{k}_{1}})$ straddling gates always suffice. We then prove any $\text{U}({2}^{n})$ unitary synthesis can be accomplished with $\mathrm{\ensuremath{\Theta}}({4}^{{k}_{1}})$ straddling gates. Furthermore, we extend our results to multipartite systems, and show that any $m$-partite Schmidt decomposable state has binding complexity linear in $m$, which hints its multiseparable property. This result not only resolves an open problem posed by Vijay Balasubramanian, who was initially motivated by the complexity = volume conjecture in quantum gravity, but also offers realistic applications in distributed quantum computation in the near future.

We prove the Turaev–Viro invariants volume conjecture for a “universal” class of cusped hyperbolic $3$-manifolds that produces all $3$-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic $3$-manifold with empty or toroidal boundary in terms of the growth rate of the Turaev–Viro invariants of the complement of an appropriate link contained in the manifold. We also provide evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum representations of surface mapping class groups. A key step in our proofs is finding a sharp upper bound on the growth rate of the quantum $6j$-symbol evaluated at $q=e\frac{2\pi i}{r}$.

The holographic complexity of formation for the AdS3 2-sided Randall-Sundrum model and the AdS3/BCFT2 models is logarithmically divergent according to the volume conjecture, while it is finite using the action proposal. One might be tempted to conclude that the UV divergences of the volume and action conjectures are always different for defects and boundaries in two-dimensional conformal field theories. We show that this is not the case. In fact, in Janus AdS3 we find that both volume and action proposals provide the same kind of logarithmic divergences.

We use geometric methods to show that given any $3$-manifold $M$, and $g$ a sufficiently large integer, the mapping class group $\mathrm{Mod}(\Sigma_{g,1})$ contains a coset of an abelian subgroup of rank $\lfloor \frac{g}{2}\rfloor,$ consisting of pseudo-Anosov monodromies of open-book decompositions in $M.$ We prove a similar result for rank two free cosets of $\mathrm{Mod}(\Sigma_{g,1}).$ These results have applications to a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. For surfaces with boundary, and large enough genus, we construct cosets of abelian and free subgroups of their mapping class groups consisting of elements that satisfy the conjecture. The mapping tori of these elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viro invariants volume conjecture.