Bottom tangles and universal invariants

Habiro, Kazuo

doi:10.2140/agt.2006.6.1113

Cited by 61 publications

(130 citation statements)

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“…In Section 4, we first recall from [20] the notion of bottom tangles. An ncomponent bottom tangle T is a tangle in a cube consisting of n arc components whose endpoints lie in a line in the bottom square of the cube in such a way that between the two endpoints of each arc there are no endpoints of other arcs.…”

Section: The Ring Z[q]mentioning

confidence: 99%

“…Bottom tangles. Here we recall from [20] the notion of bottom tangles. An n-component bottom tangle T = T 1 ∪ · · · ∪ T n is a framed tangle in a cube, which is drawn as a diagram in a rectangle as usual, consisting of n arcs T 1 , .…”

Section: Universal Sl 2 Invariant Of Bottom Tanglesmentioning

confidence: 99%

“…To prove Theorem 4.1, we use the following two results from [20]. (1) 1 ∈ X 0 , 1 ∈ X 1 , and J B ∈ X 3 .…”

Section: 2mentioning

confidence: 99%

“…We have not worked out all the details yet, but the following idea seems useful. The main tool is [20,Theorem 9.9], which reduces the problem to showing that…”

Section: 5mentioning

confidence: 99%

“…Here Y : U h⊗ U h → U h is the commutator morphism for the transmutation of U h (see [20,Section 9.3] for the definition).…”

Section: 5mentioning

confidence: 99%

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A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

Habiro

2007

Invent. math.

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238

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Abstract. We construct an invariant J M 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) Witten-ReshetikhinTuraev invariant τ ζ (M ) of M at ζ. Thus J M unifies all the SU (2) WittenReshetikhin-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 ζ behaves 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 unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τ ζ (M ) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.

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Section: The Ring Z[q]mentioning

confidence: 99%

Section: Universal Sl 2 Invariant Of Bottom Tanglesmentioning

confidence: 99%

“…To prove Theorem 4.1, we use the following two results from [20]. (1) 1 ∈ X 0 , 1 ∈ X 1 , and J B ∈ X 3 .…”

Section: 2mentioning

confidence: 99%

“…We have not worked out all the details yet, but the following idea seems useful. The main tool is [20,Theorem 9.9], which reduces the problem to showing that…”

Section: 5mentioning

confidence: 99%

“…Here Y : U h⊗ U h → U h is the commutator morphism for the transmutation of U h (see [20,Section 9.3] for the definition).…”

Section: 5mentioning

confidence: 99%

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