A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space
[Une globalisation des polynômes de Jones et d’Alexander construite à partir de l’intersection graduée de deux lagrangiens dans un espace de configurations]
Anghel, Cristina Ana-Maria1, 2
1 University of Geneva, Section de mathématiques, Rue du Conseil-Général 7-9, Geneva CH 1205, Switzerland
2 Institute of Mathematics “Simion Stoilow”, Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
Annales de l'Institut Fourier, Online first, 48 p.

Nous considérons deux polynômes de Laurent à deux variables associés à une tresse, donnés par des intersections graduées entre lagrangiens fixes dans un espace de configurations. Afin d’obtenir des invariants d’entrelacs, on remarque qu’il faut faire le quotient par une relation quadratique. Ensuite nous prouvons par des outils topologiques que cette relation est suffisante et la première intersection graduée donne un invariant qui est le polynôme de Jones. Cela montre un modèle topologique pour le polynôme de Jones et une preuve topologique directe qu’il s’agit d’un invariant bien défini. L’autre modèle d’intersection dans le quotient est un invariant qui globalise les polynômes de Jones et d’Alexander. Cette globalisation dans l’anneau quotient est donnée par une interpolation spécifique entre les polynômes d’Alexander et de Jones.

We consider two Laurent polynomials in two variables associated to a braid, given by graded intersections between fixed Lagrangians in configuration spaces. In order to get link invariants, we notice that we have to quotient by a quadratic relation. Then we prove by topological tools that this relation is sufficient and the first graded intersection gives an invariant which is the Jones polynomial. This shows a topological model for the Jones polynomial and a direct topological proof that it is a well-defined invariant. The other intersection model in the quotient turns out to be an invariant globalising the Jones and Alexander polynomials. This globalisation in the quotient ring is given by a specific interpolation between the Alexander and Jones polynomials.

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DOI : 10.5802/aif.3692
Classification : 20C08, 20C12, 20F36, 55N25, 55R80, 57M10
Keywords: Quantum invariants, Topological models, Graded intersections, Symmetric powers
Mots-clés : Invariants quantiques, Modèles topologiques, Intersections graduées, Puissances symétriques

Anghel, Cristina Ana-Maria 1, 2

1 University of Geneva, Section de mathématiques, Rue du Conseil-Général 7-9, Geneva CH 1205, Switzerland
2 Institute of Mathematics “Simion Stoilow”, Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania 
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Anghel, Cristina Ana-Maria. A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space. Annales de l'Institut Fourier, Online first, 48 p.

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