A cohomology theory for colored tangles

Carmen Caprau

Displaying similar documents to “A cohomology theory for colored tangles”

Brunnian local moves of knots and Vassiliev invariants

Akira Yasuhara (2006)

Fundamenta Mathematicae

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K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of $C_{k}$ -moves. In this paper we give another geometric condition in terms of Brunnian local moves. The proof is simple and self-contained.

Algebras of the cohomology operations in some cohomology theories

A. Jankowski

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Contents0. Introduction............................................................................................................................................. 51. Preliminaries.......................................................................................................................................... 62. Generalized cohomology theories with a coefficient group $Z_{p}$ .............................................. 83. Cohomology theory BP* ( , $Z_{p}$ )........................................................................................................

The Seiberg–Witten invariants of negative definite plumbed 3-manifolds

András Némethi (2011)

Journal of the European Mathematical Society

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Assume that $Γ$ is a connected negative definite plumbing graph, and that the associated plumbed 3-manifold $M$ is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg–Witten invariant of $M$ . The first one is the constant term of a ‘multivariable Hilbert polynomial’, it reflects in a conceptual way the structure of the graph $Γ$ , and emphasizes the subtle parallelism between these topological invariants and the analytic invariants of normal surface singularities....

A review of Lie superalgebra cohomology for pseudoforms

Carlo Alberto Cremonini (2022)

Archivum Mathematicum

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This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational...

Elementary moves for higher dimensional knots

Dennis Roseman (2004)

Fundamenta Mathematicae

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For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in $ℝ^{n + 2}$ (or $^{n + 2}$ ), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

Legendrian and transverse twist knots

John B. Etnyre, Lenhard L. Ng, Vera Vértesi (2013)

Journal of the European Mathematical Society

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In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m (5_{2})$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot $K_{- 2 n}$ with crossing number $2 n + 1$ . In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{- 2 n}$ has exactly $⌈\frac{n^{2}}{2}⌉$ Legendrian representatives with maximal Thurston–Bennequin...

Link invariants from finite racks

Sam Nelson (2014)

Fundamenta Mathematicae

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We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.

Motivic cohomology and unramified cohomology of quadrics

Bruno Kahn, R. Sujatha (2000)

Journal of the European Mathematical Society

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This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension $\leq 4$ and $\geq 11$ . Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real...

Quantum Cohomology and Crepant Resolutions: A Conjecture

Tom Coates, Yongbin Ruan (2013)

Annales de l’institut Fourier

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We give an expository account of a conjecture, developed by Coates–Iritani–Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold $𝒳$ to the quantum cohomology of a crepant resolution $Y$ of $𝒳$ . We explore some consequences of this conjecture, showing that it implies versions of both the Cohomological Crepant Resolution Conjecture and of the Crepant Resolution Conjectures of Ruan and Bryan–Graber. We also give a ‘quantized’ version of the conjecture, which determines...

$L^{2}$ -cohomology

J. Eichhorn (1986)

Banach Center Publications

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Cocycle invariants of codimension 2 embeddings of manifolds

Józef H. Przytycki, Witold Rosicki (2014)

Banach Center Publications

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We consider the classical problem of a position of n-dimensional manifold Mⁿ in $ℝ^{n + 2}$ . We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting $M ⁿ \to ℝ^{n + 2}$ . In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of Mⁿ embedded in $ℝ^{n + 2}$ we have (n+1)- and (n+2)-(co)cycle invariants (i.e. invariant under Roseman moves).

A conjecture on Khovanov's invariants

Stavros Garoufalidis (2004)

Fundamenta Mathematicae

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We formulate a conjectural formula for Khovanov's invariants of alternating knots in terms of the Jones polynomial and the signature of the knot.

Asymptotic Expansions of Witten-Reshetikhin-Turaev Invariants for Some Simple $3$ -Manifolds

R. J. Lawrence (1995)

Recherche Coopérative sur Programme n°25

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Overconvergent de Rham-Witt cohomology

Christopher Davis, Andreas Langer, Thomas Zink (2011)

Annales scientifiques de l'École Normale Supérieure

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The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p > 0$ , an overconvergent de Rham-Witt complex $W^{†} Ω_{X / k}$ as a suitable subcomplex of the de Rham-Witt complex of Deligne-Illusie. This complex, which is functorial in $X$ , is a complex of étale sheaves and a differential graded algebra over the ring $W^{†} (𝒪_{X})$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent...

On the Signatures of Torus Knots

Maciej Borodzik, Krzysztof Oleszkiewicz (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study properties of the signature function of the torus knot $T_{p, q}$ . First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.

BGG resolutions via configuration spaces

Michael Falk, Vadim Schechtman, Alexander Varchenko (2014)

Journal de l’École polytechnique — Mathématiques

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We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik–Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the ${𝔰𝔩}_{2}$ Bernstein–Gelfand–Gelfand resolution as an Aomoto complex.

On the colored Jones polynomials of ribbon links, boundary links and Brunnian links

Sakie Suzuki (2014)

Banach Center Publications

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Habiro gave principal ideals of $ℤ [q, q^{- 1}]$ in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links and Brunnian links are contained in smaller ideals of $ℤ [q, q^{- 1}]$ generated by several elements. In this paper, we prove that these ideals also are principal, each generated by a product of cyclotomic polynomials.

Asymptotic Vassiliev invariants for vector fields

Sebastian Baader, Julien Marché (2012)

Bulletin de la Société Mathématique de France

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We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $ℝ^{3}$ . More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field.

Examples on Polynomial Invariants of Knots and Links.

Taizo Kanenobu (1986)

Mathematische Annalen

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A Künneth formula in topological homology and its applications to the simplicial cohomology of $ℓ ¹ (ℤ ₊^{k})$

F. Gourdeau, Z. A. Lykova, M. C. White (2005)

Studia Mathematica

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We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(’) ≅ Hₙ()’, where Hₙ()’ is the dual space of the homology group of and Hⁿ(’) is the cohomology group of the dual complex ’. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe...

Bounded cohomology of lattices in higher rank Lie groups

Marc Burger, Nicolas Monod (1999)

Journal of the European Mathematical Society

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We prove that the natural map $H_{b}^{2} (Γ) \to H^{2} (Γ)$ from bounded to usual cohomology is injective if $Γ$ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for $Γ$ : the stable commutator length vanishes and any $C^{1}$ –action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating $H^{*} b (Γ)$ to the continuous bounded cohomology of the...

Редуцированные $L_{p}$ -когомологии искривленных цилиндров.

В.М. Гольдштейн, В.И. Кузьминов, И.А. Шведов (1990)

Sibirskij matematiceskij zurnal

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Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type $𝐀_{1}$

Bo Hou, Yanhong Guo (2015)

Czechoslovak Mathematical Journal

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The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let $Λ_{t}$ be the Yoneda algebra of a reconstruction algebra of type $𝐀_{1}$ over a field $. I n t h i s p a p e r, a m i n i m a l p r o j e c t i v e b i m o d u l e r e s o l u t i o n o f$ t $i s c o n s t r u c t e d, a n d t h e$ -dimensions of all Hochschild homology and cohomology groups of $Λ_{t}$ are calculated explicitly.