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Salvetti complex, spectral sequences and cohomology of Artin groups

Filippo Callegaro (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.

Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

Józef Przytycki (1995)

Banach Center Publications

We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics)....

Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

David M J. Calderbank, Henrik Pedersen (2000)

Annales de l'institut Fourier

We study the Jones and Tod correspondence between selfdual conformal 4 -manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3 -manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence...

Sharp edge-homotopy on spatial graphs.

Ryo Nikkuni (2005)

Revista Matemática Complutense

A sharp-move is known as an unknotting operation for knots. A self sharp-move is a sharp-move on a spatial graph where all strings in the move belong to the same spatial edge. We say that two spatial embeddings of a graph are sharp edge-homotopic if they are transformed into each other by self sharp-moves and ambient isotopies. We investigate how is the sharp edge-homotopy strong and classify all spatial theta curves completely up to sharp edge-homotopy. Moreover we mention a relationship between...

Shrinking of toroidal decomposition spaces

Daniel Kasprowski, Mark Powell (2014)

Fundamenta Mathematicae

Given a sequence of oriented links L¹,L²,L³,... each of which has a distinguished, unknotted component, there is a decomposition space 𝓓 of S³ naturally associated to it, which is constructed as the components of the intersection of an infinite sequence of nested solid tori. The Bing and Whitehead continua are simple, well known examples. We give a necessary and sufficient criterion to determine whether 𝓓 is shrinkable, generalising previous work of F. Ancel and M. Starbird and others. This criterion...

Signature of rotors

Mieczysław K. Dąbkowski, Makiko Ishiwata, Józef H. Przytycki, Akira Yasuhara (2004)

Fundamenta Mathematicae

Rotors were introduced as a generalization of mutation by Anstee, Przytycki and Rolfsen in 1987. In this paper we show that the Tristram-Levine signature is preserved by orientation-preserving rotations. Moreover, we show that any link invariant obtained from the characteristic polynomial of the Goeritz matrix, including the Murasugi-Trotter signature, is not changed by rotations. In 2001, P. Traczyk showed that the Conway polynomials of any pair of orientation-preserving rotants coincide. We show...

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