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Cocycle invariants of codimension 2 embeddings of manifolds

Józef H. Przytycki, Witold Rosicki (2014)

Banach Center Publications

We consider the classical problem of a position of n-dimensional manifold Mⁿ in 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 + 2 . In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of Mⁿ embedded in n + 2 we have (n+1)- and (n+2)-(co)cycle invariants (i.e. invariant under Roseman moves).

Combinatorial mapping-torus, branched surfaces and free group automorphisms

François Gautero (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a 1 -cocycle of a 2 -complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates...

Combinatorics and topology - François Jaeger's work in knot theory

Louis H. Kauffman (1999)

Annales de l'institut Fourier

François Jaeger found a number of beautiful connections between combinatorics and the topology of knots and links, culminating in an intricate relationship between link invariants and the Bose-Mesner algebra of an association scheme. This paper gives an introduction to this connection.

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