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Quantum invariants of 3manifolds: integrality, splitting, and perturbative expansion
 In Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of ThreeManifolds
, 1999
"... Abstract. We consider quantum invariants of 3manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that t ..."
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Abstract. We consider quantum invariants of 3manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invariant is always an algebraic integer, if the quantum parameter is a prime root of unity. We also show that the projective quantum invariant of rational homology 3spheres has a perturbative expansion a la Ohtsuki. The presentation of the theory of quantum 3manifold is selfcontained. 0.1. For a simple Lie algebra g over C with Cartan matrix (aij) let d = maxi̸=j aij. Thus d = 1 for the ADE series, d = 2 for BCF and d = 3 for G2. The quantum group associated with g is a Hopf algebra over Q(q 1/2), where q 1/2 is the quantum parameter. To fix the order let us point out that our q is q 2 in [Ka, Ki, Tu] or v 2 in the book [Lu2]. For example, the quantum
A unified WittenReshetikhinTuraev invariant for integral homology spheres
, 2006
"... We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTurae ..."
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Cited by 37 (4 self)
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We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTuraev invariants of M. As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) as a function on ζ behaves like an “analytic function ” defined on the set of roots of unity. That is, the τζ(M) for all roots of unity are determined by a “Taylor expansion ” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.
Analyticity of the free energy of a closed 3manifold
 AND ASYMPTOTICS OF GRAPH COUNTING PROBLEMS IN UNORIENTED SURFACES 23
"... Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold ..."
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Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of BenderGaoRichmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of BenderGaoRichmond.
On padic properties of the WittenReshetikhinTuraev invariant
, 1999
"... We prove the Lawrence conjecture about padic convergence of the series of Ohtsuki invariants of a rational homology sphere to its SO(3) WittenReshetikhinTuraev invariant. Our proof is based on the surgery formula for Ohtsuki series and on the properties of the expansion of the colored Jones polyn ..."
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Cited by 8 (0 self)
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We prove the Lawrence conjecture about padic convergence of the series of Ohtsuki invariants of a rational homology sphere to its SO(3) WittenReshetikhinTuraev invariant. Our proof is based on the surgery formula for Ohtsuki series and on the properties of the expansion of the colored Jones polynomial of a knot in powers of q − 1 and qα − 1, α being the color of the knot.
Almost integral TQFTs from simple Lie algebras
, 2005
"... ... aim of this paper is to modify the TQFT of the category of extended 3cobordisms given by [T] to obtain almost integral TQFT. ..."
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Cited by 5 (4 self)
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... aim of this paper is to modify the TQFT of the category of extended 3cobordisms given by [T] to obtain almost integral TQFT.
A UNIFIED QUANTUM SO(3) INVARIANT FOR RATIONAL HOMOLOGY 3–SPHERES
, 801
"... Abstract. Given a rational homology 3–sphere M with H1(M, Z)  = b and a link L inside M, colored by odd numbers, we construct a unified invariant IM,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten–Reshetikhin– ..."
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Cited by 4 (3 self)
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Abstract. Given a rational homology 3–sphere M with H1(M, Z)  = b and a link L inside M, colored by odd numbers, we construct a unified invariant IM,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten–Reshetikhin–Turaev invariants of the pair (M, L). If b = 1 and L = ∅, IM coincides with Habiro’s invariant of integral homology 3–spheres. For b> 1, the unified invariant defined by the third author is determined by IM. One of the applications are the new Ohtsuki series (perturbative expansions of IM at roots of unity) dominating all quantum SO(3) invariants.
Unified quantum invariants and their refinements for homology 3spheres with 2torsion, Fund
 Math
"... Abstract. For every rational homology 3–sphere with H1(M, Z) = (Z/2Z) n we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten–Reshetikhin–Turaev inv ..."
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Abstract. For every rational homology 3–sphere with H1(M, Z) = (Z/2Z) n we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten–Reshetikhin–Turaev invariant at this root and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invariant splits into a sum of the refined unified invariants dominating spin and cohomological refinements of quantum SU(2) invariants. New results on the Ohtsuki series and the integrality of quantum invariants are the main applications of our construction.
Quantum SU(2)–invariants for three–manifolds associated with non– trivial cohomology classes modulo two
 Knots in Hellas ’98, Proceed. Internat. Conference on Knot Theory and its Ramif., Series of Knots and Everything 24 (2000) 347–352 (http://www3.tky.3web.ne.jp/%7Estarshea/paper/hellas.ps.zip
"... Abstract. We show an integrality of the quantum SU(2)invariant associated with a nontrivial first cohomology class modulo two. For a given Lie group G and an integer (level) k, E. Witten gave an ‘invariant ’ of threemanifolds by using the path integral [33]. Mathematically rigorous proof of its e ..."
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Abstract. We show an integrality of the quantum SU(2)invariant associated with a nontrivial first cohomology class modulo two. For a given Lie group G and an integer (level) k, E. Witten gave an ‘invariant ’ of threemanifolds by using the path integral [33]. Mathematically rigorous proof of its existence was given by several people. See for example [26, 7, 4, 32, 29]. There are also some refinements for these invariants corresponding to various structures of the threemanifolds, such as cohomology classes and spin structures [30, 7, 9, 18, 3, 2]. Some topological properties of the invariants were given especially for G = SU(2). The first striking result was found by R. Kirby and P. Melvin stating that the quantum SU(2)invariant with level two splits into the sum of the invariants associated with spin structures and each summand can be described in terms of the µinvariant. They and X. Zhang obtained a topological interpretation for the quantum SU(2)invariant with level four [8]. In this case the invariant splits into the sum of the invariants associated with the first cohomology classes modulo two and each summand can be described in terms of the Witt invariant and
Hyperbolic threemanifolds with trivial finite type invariants
 Kobe J. Math
, 1999
"... Abstract. We construct a hyperbolic threemanifold with trivial finite type invariants up to a given degree. The concept of finite type invariants for integral homology threespheres was introduced by T. Ohtsuki in [10] including the Casson invariant [1] for the first nontrivial example. It attract ..."
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Abstract. We construct a hyperbolic threemanifold with trivial finite type invariants up to a given degree. The concept of finite type invariants for integral homology threespheres was introduced by T. Ohtsuki in [10] including the Casson invariant [1] for the first nontrivial example. It attracts not only mathematicians but also physicists since it is closely related to E. Witten’s quantum invariants for threemanifolds [13] in the following way. It is proved by T.T.Q. Le [5] that the degree d term of the LMO invariant [6] is of finite type of degree d and conversely any finite type invariant of degree d comes from the degree d term of the LMO invariant. Since it is also known that the perturbative PSU(n) invariant can be obtained from the LMO invariant [8, 4], every coefficient of the perturbative PSU(n) invariant is of finite type of degree d. (For n = 2 case was also proved in [3].) Therefore we may say that finite type invariants approximate quantum invariants from lower degree parts. We refer the reader to [9, 7] for more detail. It is a natural question to ask how strong finite type invariants are. In this paper we give at least one threemanifold with trivial finite type invariants up to a given degree. Acknowledgment. This work was done when the author was visiting the MittagLeffler Institute. He thanks the staffs for their hospitality: ‘Tack s˚a mycket’. He also thanks Kazuo Habiro and Tomotada Ohtsuki for useful conversations. Thanks are also due to the subscribers to the mailing list ‘knot ’
LAPLACE TRANSFORM AND UNIVERSAL sl2 INVARIANTS
, 2005
"... Abstract. We develop a Laplace transform method for constructing universal invariants of 3–manifolds. As an application, we recover Habiro’s theory of integer homology 3–spheres and extend it to some classes of rational homology 3–spheres with cyclic homology. If H1  = 2, we give explicit formula ..."
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Abstract. We develop a Laplace transform method for constructing universal invariants of 3–manifolds. As an application, we recover Habiro’s theory of integer homology 3–spheres and extend it to some classes of rational homology 3–spheres with cyclic homology. If H1  = 2, we give explicit formulas for universal invariants dominating the sl2 and SO(3) Witten–Reshetikhin–Turaev invariants, as well as their spin and cohomological refinements at all roots of unity. New results on the Ohtsuki series and the integrality of quantum invariants are the main applications of our construction.