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

Kuriya, Takahito; Le, Thang T. Q.; Ohtsuki, Tatsuyuki

doi:10.1112/jtopol/jts010

Cited by 5 publications

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“…For homology 3-spheres, one can construct infinite series of finite-type invariants following Ohtsuki's original idea [83], by appropriate expansions of quantum invariants. Furthermore, there is a very powerful invariant of homology 3-spheres: the LMO invariant [53], which is known to be universal among Q-valued finite-type invariants [52] and to dominate large families of quantum invariants [47]. For homology cylinders too, there is a universal Q-valued finite-type invariant: the LMO homomorphism defined on the monoid IC(), which allows for an explicit diagrammatic description of the Lie algebra Gr Y IC() with rational coefficients [9,31].…”

Section: I-32mentioning

confidence: 99%

Surgery equivalence relations for 3-manifolds

Massuyeau

2024

Winter Braids Lecture Notes

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Section: I-32mentioning

confidence: 99%

Surgery equivalence relations for 3-manifolds

Massuyeau

2024

Winter Braids Lecture Notes

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“…. As explained in [KLO12], this LMO invariant contains the quantum Witten-Reshetikhin-invariants of rational homology 3-spheres defined in [RT91].…”

Section: Sketch Of Proofmentioning

confidence: 99%

An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations

Lescop

2013

Preprint

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The finite type invariant concept for knots was introduced in the 90's in order to classify knot invariants, with the work of Vassiliev, Goussarov and Bar-Natan, shortly after the birth of numerous quantum knot invariants. This very useful concept was extended to 3-manifold invariants by Ohtsuki.These introductory lectures show how to define finite type invariants of links and 3manifolds by counting graph configurations in 3-manifolds, following ideas of Witten and Kontsevich.The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. As an important example, we present it as the algebraic intersection of a torus and a 4-chain called a propagator in a configuration space.In the second section, we introduce the simplest finite type 3-manifold invariant, which is the Casson invariant (or the Θ-invariant) of integer homology 3-spheres. It is defined as the algebraic intersection of three propagators in the same two-point configuration space.In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams.In Sections 4 and 5, we sketch an original construction based on configuration space integrals of universal finite type invariants for links in rational homology 3-spheres and we state open problems. Our construction generalizes the known constructions for links in R 3 and for rational homology 3-spheres, and it makes them more flexible.In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.

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An extension of the LMO functor

Nozaki¹

2016

Geom Dedicata

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Abstract. Cheptea, Habiro and Massuyeau constructed the LMO functor, which is defined on a certain category of cobordisms between two surfaces with at most one boundary component. In this paper, we extend the LMO functor to the case of any number of boundary components, and our functor reflects relations among the parts corresponding to the genera and boundary components of surfaces. We also discuss a relationship with finite-type invariants and Milnor invariants.

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

Cited by 5 publications

References 35 publications

Surgery equivalence relations for 3-manifolds

Surgery equivalence relations for 3-manifolds

An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations

An extension of the LMO functor

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