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38
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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Cited by 35 (9 self)
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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
INTEGRALITY AND SYMMETRY OF QUANTUM LINK Invariants
 VOL. 102, NO. 2 DUKE MATHEMATICAL JOURNAL
, 2000
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Integrality for TQFTs
 Duke Math J
"... Abstract. We discuss ways that the ring of coefficients for a TQFT can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free f ..."
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Cited by 20 (9 self)
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Abstract. We discuss ways that the ring of coefficients for a TQFT can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers: Z[ζ2p], if p ≡ −1 (mod 4), and Z[ζ4p], if p ≡ 1 (mod 4), where ζk is a primitive kth root of unity. We study the quantum invariants of prime power order simple cyclic covers of 3manifolds. We define new invariants arising from strong shift equivalence and integrality. Similar results are obtained for some other TQFTs but the modules are only guaranteed to be projective.
Irreducibility of some quantum representations of mapping class groups
 J. Knot Theory Ramifications
"... Abstract. The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given. 1. ..."
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Cited by 18 (0 self)
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Abstract. The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given. 1.
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 17 (2 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.
On the WittenReshetikhinTuraev representations of the mapping class groups
 Proc. Amer. Math. Soc
, 1999
"... We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The WittenReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representa ..."
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Cited by 14 (5 self)
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We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The WittenReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus with no colored points, the representations have finite image. Recall that an object in a cobordism category of dimension 2+1 is a closed oriented surfaces Σ perhaps with some specified further structure. A morphism M from Σ to Σ ′ is (loosely speaking) a compact oriented 3manifold perhaps with some specified further structure, called a cobordism, whose boundary is the disjoint union of −Σ and Σ ′. A morphism M ′ from Σ ′ to Σ ′ ′ is composed with a morphism from Σ to Σ ′ by gluing along Σ ′ , inducing any required extra structure from the structures on M and M ′. Also the extra structure on a 3manifold must induce the extra structure on the boundary. A TQFT in dimension 2+1 is then a functor from
Quantum cyclotomic orders of 3manifolds
 Topology
"... Abstract. This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3manifolds. In particular, it is shown that the padic valuation of the quantum SO(3)invariant of a 3manifold M, for odd primes p, is bounded below by a linear function of the mod p ..."
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Cited by 12 (1 self)
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Abstract. This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3manifolds. In particular, it is shown that the padic valuation of the quantum SO(3)invariant of a 3manifold M, for odd primes p, is bounded below by a linear function of the mod p first betti number of M. Sharper bounds using more delicate topological invariants are given as well. Since the birth of quantum topology in the last decade [Jo][Wi], one of the fundamental problems facing topologists has been to find topological interpretations for the vast array of quantum invariants that have come to light. One common characteristic among these invariants is their rich number theoretic content, and it has been a
3Manifold invariants and periodicity of homology spheres
, 2001
"... We show how the periodicity of a homology sphere is reflected in the ReshetikhinTuraevWitten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere. ..."
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Cited by 10 (1 self)
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We show how the periodicity of a homology sphere is reflected in the ReshetikhinTuraevWitten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.
STRONG INTEGRALITY OF QUANTUM INVARIANTS OF 3MANIFOLDS
"... Abstract. We prove that the quantum SO(3)invariant of an arbitrary 3manifold M is always an algebraic integer if the order of the quantum parameter is coprime with the order of the torsion part of H1(M,Z). An even stronger integrality, known as cyclotomic integrality, was established by Habiro fo ..."
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Cited by 9 (4 self)
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Abstract. We prove that the quantum SO(3)invariant of an arbitrary 3manifold M is always an algebraic integer if the order of the quantum parameter is coprime with the order of the torsion part of H1(M,Z). An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3spheres. Here we also generalize Habiro’s result to all rational homology 3spheres. 0.1. Integrality at roots of nonprime order. Let τM(q) be the quantum SO(3)invariant of a 3manifold M, which can be defined when q is a complex root of unity of order odd and greater than 1. The quantum SU(2)invariant was defined by Reshetikhin and Turaev (see [Tur]) and the SO(3)version was defined