Quantum Invariants of Knots and 3-Manifolds

Turaev, Vladimir

doi:10.1515/9783110221848

Cited by 700 publications

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“…The left duality maps (A.1) are respectively depicted by the first two ribbon graphs in Figure 4, and the twist by the last ribbon graph in Figure 4, where again the directions of the cores of ribbons are understood to be consistent with sources and targets of the ribbon graphs. The category H (K) can be presented [38,47] in terms of the generators depicted in Figure 1, and the relations given in, e.g., [38,47]. Figure 5.…”

Section: 4mentioning

confidence: 99%

“…The following theorem is a special case of the tensor functor [38,43,47] from the category of coloured ribbon graphs to any given ribbon category. The general result may be found in, e.g., [47, Theorem 2.5].…”

Section: 4mentioning

confidence: 99%

“…The full set of relations can be obtained from those given in [38,47] for the category of ribbon graphs by ignoring the directions of the cores of ribbons.…”

Section: 4mentioning

confidence: 99%

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The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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Abstract. Let U q (g) be the quantum supergroup of gl m|n or the modified quantum supergroup of osp m|2n over the field of rational functions in q, and let V q be the natural module for U q (g). There exists a unique tensor functor, associated with V q , from the category of ribbon graphs to the category of finite dimensional representations of U q (g), which preserves ribbon category structures. We show that this functor is full in the cases g = gl m|n or osp 2ℓ+1|2n . For g = osp 2ℓ|2n , we show that the space Hom Uq(g) (V ⊗r q , V ⊗s q ) is spanned by images of ribbon graphs if r + s < 2ℓ(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g).

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Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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“…As the quantum invariant I(M) provides a lower bound on Heegaard genus of M ( [14]), this result implies that for any genus g, there are infinitely many three-manifolds with Heegaard genus at least g.…”

Section: Introductionmentioning

confidence: 86%

“…We establish notation and briefly review the pertinent SO(3) quantum theory associated to r and A. We refer the reader to either Turaev's book ( [14]) or Lickorish's survey ( [8]) for details on the basic constructions.…”

Section: So(3) Quantum Theorymentioning

confidence: 99%

SO(3) quantum invariants are dense

Wong¹

2009

Math. Proc. Camb. Phil. Soc.

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Boundaries, defects and Frobenius algebras

Fuchs¹,

Runkel²,

Schweigert³

2003

Fortschritte der Physik

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The interpretation of D-branes in terms of open strings has lead to much interest in boundary conditions of two-dimensional conformal field theories (CFTs). These studies have deepened our understanding of CFT and allowed us to develop new computational tools. The construction of CFT correlators based on combining tools from topological field theory and non-commutative algebra in tensor categories, which we summarize in this contribution, allows e.g. to discuss, apart from boundary conditions, also defect lines and disorder fields.

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Quantum Invariants of Knots and 3-Manifolds

Cited by 700 publications

References 0 publications

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

SO(3) quantum invariants are dense

Boundaries, defects and Frobenius algebras

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