A Combinatorial Description of Knotted Surfaces and Their Isotopies

Carter, J.Scott; Rieger, J. H.; Saito, Masahico

doi:10.1006/aima.1997.1618

Cited by 42 publications

(71 citation statements)

References 18 publications

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“…However, the representation theoretic problem for su 2 does not exist for su 3 , whose fundamental representations are dual to each other, but not self-dual. This, along with some experimental evidence, led Morrison and Nieh to conjecture that the su 3 theory is honestly functorial, with no sign anomaly.…”

Section: Resultsmentioning

confidence: 99%

Functoriality for the su₃Khovanov homology

Clark

2009

Algebr. Geom. Topol.

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We prove that the categorified su 3 quantum link invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su 2 theory [8; 1], which was not functorial as originally defined [7; 5].We use methods of Morrison and Nieh [12] and Bar-Natan [2] to construct explicit chain maps for each variation of the third Reidemeister move. Then, to show functoriality, we modify arguments used by Clark, Morrison and Walker [5] to show that induced chain maps are invariant, up to homotopy, under Carter and Saito's movie moves [4; 3].

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Section: Resultsmentioning

confidence: 99%

Functoriality for the su₃Khovanov homology

Clark

2009

Algebr. Geom. Topol.

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“…6 He called it a "canopoly", instead, but we're taking the liberty of fixing the name. 7 See also Webster [22] for a description of Khovanov-Rozansky homology [16; 17] using canopolises.…”

Section: A4 Planar Algebras and Canopolisesmentioning

confidence: 99%

Fixing the functoriality of Khovanov homology

2009

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We describe a modification of Khovanov homology [13], in the spirit of Bar-Natan [2], which makes the theory properly functorial with respect to link cobordisms. This requires introducing "disorientations" in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations have "seams" separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation).We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter, Rieger and Saito's movie moves [8; 7] always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a "local" result about tangles. Along the way, we reproduce Jacobsson's sign table [10] for the original "unoriented theory", with a few disagreements.

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“…Handle decompositions of complements of knotted surfaces. For details on knotted surfaces, in particular the movie presentation of them, we refer the reader to [14,13,12,21]. We work in the smooth category.…”

Section: Complements Of Knotted Surfacesmentioning

confidence: 99%

“…In this case, for calculation purposes, all strands of a knot with bands K can pass through each other, except when one is a thin component and the other is a band. In addition, the coloring of an arc of K − depends only on the component of K − to which it belongs; see Figures 12,13,15, and 17, and adapt them to the case when G is abelian and ∂ = 1 G . Remark 3.6.…”

Section: 21mentioning

confidence: 99%

The fundamental crossed module of the complement of a knotted surface

Martins

2009

Trans. Amer. Math. Soc.

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Abstract. We prove that if M is a CW-complex and M 1 is its 1-skeleton, then the crossed module Π 2 (M, M 1 ) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π 2 (D 2 , S 1 ). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π 2 (M, M 1 ) → G can be re-scaled to a homotopy invariant I G (M ), depending only on the algebraic 2-type of M . We describe an algorithm for calculating π 2 (M, M (1) ) as a crossed module over π 1 (M (1) ), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0-and 1-handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant I G yields a non-trivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.

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A Combinatorial Description of Knotted Surfaces and Their Isotopies

Cited by 42 publications

References 18 publications

Functoriality for the su₃Khovanov homology

Functoriality for the su₃Khovanov homology

Fixing the functoriality of Khovanov homology

The fundamental crossed module of the complement of a knotted surface

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A Combinatorial Description of Knotted Surfaces and Their Isotopies

Cited by 42 publications

References 18 publications

Functoriality for the su3Khovanov homology

Functoriality for the su3Khovanov homology

Fixing the functoriality of Khovanov homology

The fundamental crossed module of the complement of a knotted surface

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Functoriality for the su₃Khovanov homology

Functoriality for the su₃Khovanov homology