Combinatorial cohomology of the space of long knots

Mortier, Arnaud¹

¹15 Route des Futaies, 57100 Thionville, France, Osaka City University Advanced Mathematical Institute, OCAMI, 3-3-138 Sugimotocho, Sumiyoshi-ku, Osaka-shi, Osaka 558-8585, Japan

Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3435-3465

Résumé

The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain complex such that the elements of an explicit submodule in the cohomology define algebraic intersections with some “geometrically simple” strata in the space of knots. Such strata are endowed with explicit co-orientations that are canonical in some sense. The combinatorial tools involved are natural generalisations (degeneracies) of usual methods using arrow diagrams.

Détail

DOI : 10.2140/agt.2015.15.3435

Keywords: space of knots, cohomology, Gauss diagram, arrow diagram, finite type, Vassiliev, Teiblum–Turchin, quadrisecant

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