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.
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.
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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
CC-BY-ND 4.0
@article{AIF_2025__75_6_2609_0,
author = {Anghel, Cristina Ana-Maria},
title = {A globalisation of {Jones} and {Alexander} polynomials constructed from a graded intersection of two {Lagrangians} in a configuration space},
journal = {Annales de l'Institut Fourier},
pages = {2609--2656},
year = {2025},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {75},
number = {6},
doi = {10.5802/aif.3692},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3692/}
}
TY - JOUR AU - Anghel, Cristina Ana-Maria TI - A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space JO - Annales de l'Institut Fourier PY - 2025 SP - 2609 EP - 2656 VL - 75 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3692/ DO - 10.5802/aif.3692 LA - en ID - AIF_2025__75_6_2609_0 ER -
%0 Journal Article %A Anghel, Cristina Ana-Maria %T A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space %J Annales de l'Institut Fourier %D 2025 %P 2609-2656 %V 75 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3692/ %R 10.5802/aif.3692 %G en %F AIF_2025__75_6_2609_0
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, Volume 75 (2025) no. 6, pp. 2609-2656. doi: 10.5802/aif.3692
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