Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces

2020/02/21 by Cristina Ana-Maria Anghel, Anghel, Cristina Ana-Maria
Mathematics · #16T05 #16T25 #20F36 #57M25 #57M27 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2002.09390

openalex publication_date 2020/02/21 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by Lagrangian submanifolds. The main result proves that the N^th coloured Jones polynomial and N^th coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in each case. The first corollary explains Bigelow's picture for the Jones polynomial with noodles and forks from the quantum point of view. Secondly, we conclude that the N^th coloured Alexander polynomial is a graded intersection pairing in a \mathbb Z ⊕ \mathbb Z_N-covering of the configuration space in the punctured disc.

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