We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $\exp({2π\sqrt{-1}}/{r})$ instead of the standard $\exp({π\sqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $\infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten's Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin-Turaev invariants may have a geometric interpretation other than the original one via $SU(2)$ Chern-Simons gauge theory.
Costantino--Geer--Patureau-Mirand proved relations between the Reshetikhin--Turaev link invariants and the re-normalized link invariants for knots. Their theorem says that residues of the re-normalized link invariants are given by the Reshetikhin--Turaev link invariants. However, in the case of links, the residues are vanish, so we can not obtain any relations from the residues. In this paper, we prove that the Reshetikhin--Turaev link invariants appear in higher order terms of the re-normalized link invariants for links which are plumbed.
We show that for every spherical category $\C$ with invertible dimension, the Turaev-Viro TQFT admits a splitting into blocks which come from an HQFT, called the Turaev-Viro HQFT. The Turaev-Viro HQFT has the classifying space $B\grad$ as target space, where $\grad$ is a group obtained from the category $\C$. This construction gives a reformulation of the Turaev-Viro TQFT in terms of HQFT. Furthermore the Turaev-Viro HQFT is an extension of the \emph{homotopical Turaev-Viro invariant} which splits the Turaev-Viro invariant. An application of this result is a description of the homological twis…
We prove the Volume Conjecture for the relative Reshetikhin-Turaev invariants proposed in [29] for all pairs (M,K) such that M\K is homeomorphic to the complement of the figure-8 knot in S^3 with almost all possible cone angles.
In this paper, we examine Kitaev's lattice model for an arbitrary complex, semisimple Hopf algebra. We prove that this model gives the same topological invariants as Turaev-Viro theory. Using the description of Turaev-Viro theory as an extended TQFT, we prove that the excited states of the Kitaev model correspond to Turaev-Viro theory on a surface with boundary.
A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $Δ$-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where …
We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.
We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem not to be predicted by nei…
In a previous work arXiv:0903.4512, we have built an homotopical Turaev-Viro invariant and an HQFT from the universal graduation of a spherical category. In the present paper, we show that every graduation $(G,p)$ of a spherical category $\C$ defines an homotopical Turaev-Viro invariant $HTV_{\C}^{(G,p)}$ and an HQFT $\m{H}_{\C}^{(G,p)}$. Furthermore we show that the Turaev-Viro TQFT will be split into blocks coming the HQFT $\m{H}_{\C}^{(G,p)}$. We show that this decomposition is maximal for the universal graduation of the category, which means that for every graduation $(G,p)$ the HQFT $\m{H…
In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $Φ(q; N)$ whose special values at roots of unity are identified with the Witten-Reshetikhin-Turaev invariants of the Seifert loop for the integral homology sphere. Second, we show that the function $Φ(q; N)$ satisfies a $q$-difference equation whose classical limit coincides with a component of the character varieties of the Seifert loop. Third, we give an interpretation of the function $Φ(q; N)$ from the view point of the resurgent analysis.
This paper presents a way to estimate the Heegaard genus of a $3$-manifold using the Turaev-Viro state sum TQFT. The Turaev-Viro state sum TQFT is derived from the modular category associated to the quantum group $U_q(\mathfrak{sl}_2)$, which is unitary for some $q$ by Wenzl. Hence by Turaev and Virelizier the corresponding TQFT is unitary. We modify a proof by Garoufalidis to give a lower bound of the Heegaard genus using a unitary TQFT, and then use the software Regina to provide some known counterexamples to the rank versus genus conjecture.
In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category $C$, to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center $Z(C)$. In the present paper we give a partial proof of this statement.
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. This quantum trace satisfies conditions that render the link evaluation invariant under Kirby moves. If H is a suitable finite-dimensional Hopf algebra (not weak), our invariant reduces to the Kauffman-Radford invariant for the dual of H. If H is the Weak Hopf Algebra Tannaka-Krein …
On the asymptotic expansion of quantum invariants related to surgeries of Whitehead link I: Relative Reshetikhin-Turaev invariants and the Turaev-Viro invariants at $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{2}}}$
We give a categorical setting in which Penrose graphical calculus naturally extends to graphs drawn on the boundary of a handlebody. We use it to introduce invariants of 3-manifolds presented by Heegaard splittings. We recover Kuperberg invariants when the category comes from an involutory Hopf algebra and Turaev-Viro invariants when the category is semi-simple and spherical.
We prove that the SU(2) and SO(3) Witten-Reshetikhin-Turaev invariants of any 3-manifold with any colored link inside at any root of unity are algebraic integers.
The Chen-Yang volume conjecture states that the growth rate of the Turaev-Viro invariants of a compact oriented $3$-manifold determines its simplicial volume. In this paper we prove that the Chen-Yang conjecture is stable under $(2n+1,2)$-cabling.
The center Z(C) of a spherical fusion category C (over an arbitrary commutative ring) is modular. We give an algorithm for computing the Reshetikhin-Turaev invariant defined with Z(C). It is based on Hopf diagrams and an explicit description of the structure of the coend of Z(C).
We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simply-connected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, …
In this article, we discuss a (2+1)-dimensional topological quantum field theory, for short TQFT, with a Verlinde basis. As a conclusion of this general theory, we have a Dehn surgery formula. We show that Turaev-Viro-Ocneanu TQFT has a Verlinde basis. Several applications of this theorem are exposed. Based on Izumi's data of subfactors, we list several computations of Turaev-Viro-Ocneanu invariants for some 3-manifolds.
Gukov-Pei-Putrov-Vafa constructed $q$-series invariants called homological blocks in a physical way in order to categorify Witten-Reshetikhin-Turaev (WRT) invariants and conjectured that radial limits of homological blocks are WRT invariants. In this paper, we prove their conjecture for unimodular H-graphs. As a consequence, it turns out that the WRT invariants of H-graphs yield quantum modular forms of depth two and of weight one with the quantum set $\mathbb{Q}$. In the course of the proof of our main theorem, we first write the invariants as finite sums of rational functions. We second carr…
We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the coloured Jones polynomial. In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighbourhood of the 2-complex and the signature of its complement. Some examples are compu…
We study the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants proposed in \cite{WY4} for all pairs $(M,L)$ satisfying the property that $M\setminus L$ is homeomorphic to some fundamental shadow link complement. The hyperbolic cone structure of such $(M,L)$ can be described by using the logarithmic holonomies of the meridians of some fundamental shadow link. We show that when the logarithmic holonomies are sufficiently small and all cone angles are less than $π$, the asymptotic expansion conjecture of $(M,L)$ is true. Especially, we verify the asymptotic expansion c…
We derive an explicit formula for the Witten-Reshetikhin-Turaev SO(3)-invariants of lens spaces. We use the representation of the mapping class group of the torus corresponding to the Witten-Reshetikhin-Turaev SO(3)-TQFT to give such formula.
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 …
In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjugate linear isomorphism preserving the products of the two algebras and commuting with the actions of SL(2,Z). Via this correspondence and the Dehn surgery formula, we compute Turaev-Viro-Ocneanu invariants from several subfactors for basic 3-manifolds including lens spaces and Bri…
We state a simple criterion to prove the infiniteness of the image of Reshetikhin-Turaev irreducible representations of the mapping class groups of surfaces. We use it to study some of the Reshetikhin-Turaev representations associated to the tori with one and two punctures and derive an alternative proof of a result of Funar.
Graded spherical skein 2+1-G-HQFT and modified Turaev-Viro invariants
Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Utah State University（犹他州立大学）；Univ Bretagne Sud, CNRS UMR 6205, LMBA（布列塔尼南部大学，法国国家科学研究中心联合研究单位6205，LMBA）
本文提出基于 G 分级的染色映射和 skein 模块，用于定义 2+1-G-HQFT，并恢复修改后的 Turaev-Viro 三维不变量。
We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L), where M is a closed oriented 3-manifold and L is an oriented link in M, by a triangulation of M such that each component of L is an edge. We analyze some basic properties of these invariants, including the behavior under connected sums of pairs away and along links. These properties allow us to provide examples of links in the three-sphere having the same HOM…
By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated to K. Following Viro, we consider the endomorphisms that one obtains after coloring the meridian and longitude of the knot. We show that the traces of these endomorphisms encode the same information as the colored Jones polynomials of K at a root of unity. Most of the discussion…
We concretely construct a 2-categorically extended TQFT that extends the Reshetikhin-Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov-Voevodsky 2-vector spaces.
The Witten-Reshetikhin-Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten-Reshetikhin-Turaev invariant and trivial 3-manifold fundamental group.
Turaev's shadow formula calculates the SU(2)-Reshetikhin-Turaev-Witten invariants using shadows, and its expression is somehow similar to a Euler characteristic. We give a short proof of this formula using skein theory. The formula applies to pairs (M,G) where M is a closed oriented 3-manifold and GcM is a (possibly empty) colored framed trivalent graph (for instance, a framed knot or link).
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 th…
For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible and we compute its classical shadow, in the sense of earlier work of the authors (arXiv:1206.1638).
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer $r \geq 3$. We resolve the question of complexity for $r=3$ and $r=4$, giving simple proofs that computing Turaev-Viro invariants for $r=3$ is polynomial time, but for $r=4$ is \#P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary $r$, and show through concrete implementation and…
We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are also presented.
We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754). We use these projectors to give a categorification of the sl_3 Reshetikhin-Turaev invariant of framed tangles.
We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity ζgives the the SU(2) Witten-Reshetikhin-Turaev invariant τ_ζ(M) of M at ζ. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of M. As a consequence, τ_ζ(M) is an algebraic integer. Moreover, it follows that τ_ζ(M) as a function on ζbehaves like an ``analytic function'' defined on the set of roots of unity. That is, the τ_ζ(M) for all roots of unity are determined by a "Taylor expansion" at any root of un…
We calculate the Witte-Reshetikhi-Turaev invariant for a knot in the lens space of type L(m,1) for the N-th root of unity, and study its asymptotic behavior for large N.
In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $1/2$ is strictly less than those…
We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of links in L(p,q) as a function of the level r-2. They are given by 1 over the square root of r times one of p Laurent polynomials evaluated at e to the 2 pi i divided by 4pr. The congruence class of r modulo p determines which polynomial is applicable. If p is zero modulo four, the meridian of L(p,q) is non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2) invariants. On the other hand, we show that one may recover the element in the Kauffman bracket skein module of L(p,q) represented by a link …
