A unified quantum SO(3) invariant for rational homology 3-spheres

Beliakova, Anna; Bühler, Irmgard; Le, Thang T. Q.

doi:10.1007/s00222-010-0304-5

Cited by 10 publications

(20 citation statements)

References 24 publications

(95 reference statements)

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“…In this case we have to replace the surgery link L of M by L ∪ L ′ in all definitions, fix colors on L ′ and sum over all colorings of L only. We omit here the precise definition and refer to [2] for more details.…”

Section: Definition Of the Wrt Invariantsmentioning

confidence: 99%

“…where the existence of the unified invariant of the lens space I L(b i ,1) ∈ R b , with ev ξ (I L(b i ,1) ) = τ ′ L(b i ,1) (ξ) can be shown by a direct computation (we refer to [2] for more details). Thus if we define…”

Section: Laplace Transformmentioning

confidence: 99%

“…Since unified invariants belong to cyclotomic completions of polynomial rings, we outline their construction and the main properties. For simplicity, only the case b is a power of a prime is considered, the general case is treated in [2].…”

Section: Cyclotomic Completions Of Polynomial Ringsmentioning

confidence: 99%

“…Unified invariants of rational homology 3-spheres. In [2], we give a full generalization of the Habiro theory to rational homology 3-spheres. This requires completely new techniques coming from number theory, commutative algebra, quantum group and knot theory.…”

Section: Introductionmentioning

confidence: 99%

“…In [4,3], we showed that τ G M (ξ) for any 3-manifold M and any root of unity is always an algebraic integer. Here G = SO(3) or SU (2). The integrality of the spin and cohomological refinements is work in progress.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the Unification of Quantum 3–manifold Invariants

Beliakova¹,

Le²,

Jablan³

et al. 2011

Introductory Lectures on Knot Theory

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Abstract. In 2006 Habiro initiated a construction of generating functions for WittenReshetikhin-Turaev (WRT) invariants known as unified WRT invariants. In a series of papers together with Irmgard Bühler and Christian Blanchet we extended his construction to a larger class of 3-manifolds. The unified invariants provide a strong tool to study properties of the whole collection of WRT invariants, e.g. their integrality, and hence, their categorification.In this paper we give a survey on ideas and techniques used in the construction of the unified invariants.

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Section: Definition Of the Wrt Invariantsmentioning

confidence: 99%

Section: Laplace Transformmentioning

confidence: 99%

Section: Cyclotomic Completions Of Polynomial Ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Unification of Quantum 3–manifold Invariants

Beliakova¹,

Le²,

Jablan³

et al. 2011

Introductory Lectures on Knot Theory

Self Cite

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Cyclotomic expansions for $$\mathfrak {gl}_N$$ link invariants via interpolation Macdonald polynomials

Beliakova,

Gorsky

2024

Sel. Math. New Ser.

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In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $$\mathfrak {gl}_N$$ gl N in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the $$\mathfrak {gl}_N$$ gl N Reshetikhin–Turaev link invariants and the universal $$\mathfrak {gl}_N$$ gl N knot invariant; 2) an explicit construction of the unified $$\mathfrak {gl}_N$$ gl N invariants for integral homology 3-spheres using universal Kirby colors. These results generalize those of Habiro for $$\mathfrak {sl}_2$$ sl 2 . In addition, we give a simple proof of the fact that the universal $$\mathfrak {gl}_N$$ gl N invariant of any evenly framed link and the universal $$\mathfrak {sl}_N$$ sl N invariant of any 0-framed algebraically split link are $$\Gamma $$ Γ -invariant, where $$\Gamma =Y/2Y$$ Γ = Y / 2 Y with the root lattice Y.

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The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariant

Kuriya

Ohtsuki

2012

Journal of Topology

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We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra g, i.e., the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate the quantum invariants of integral homology 3-spheres [13,14,15], this implies that the LMO invariant dominates the quantum invariants of integral homology 3-spheres.

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A unified quantum SO(3) invariant for rational homology 3-spheres

Cited by 10 publications

References 24 publications

On the Unification of Quantum 3–manifold Invariants

On the Unification of Quantum 3–manifold Invariants

Cyclotomic expansions for $$\mathfrak {gl}_N$$ link invariants via interpolation Macdonald polynomials

The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariant

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