Combinatorial cohomology of the space of long knots

Mortier, Arnaud

doi:10.2140/agt.2015.15.3435

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Combinatorial cohomology of the space of long knots

Arnaud Mortier

Algebraic & Geometric Topology 15 (2015) 3435–3465

DOI: 10.2140/agt.2015.15.3435

Abstract

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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 55N33, 57N80

References

Publication

Received: 10 September 2014

Revised: 12 March 2015

Accepted: 21 March 2015

Published: 12 January 2016

Authors

Arnaud Mortier
15 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
http://sites.google.com/site/mortier2x0/

Volume 15, issue 6 (2015)