A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space

2022/05/16 by Anghel, Cristina Ana-Maria
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2205.07842

Abstract

We consider two Laurent polynomials in two variables associated to a braid, given by \em graded intersections between \em 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 \em topological model for the Jones polynomial and a direct \em topological proof\hspace0.4mm 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 \em specific interpolation between the Alexander and Jones polynomials.

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