Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariants

TL;DR

This paper explores the mathematical relationships between Poincare series, Coxeter polynomials, and invariants of links, introducing new formulas and conjectures that connect algebraic, geometric, and topological aspects of singularities and group theory.

Contribution

It introduces generalized formulas and decompositions for Poincare series and Coxeter polynomials, linking them to the Burau representation and Milnor invariants, and formulates a conjecture connecting these to singularity theory.

Findings

Derived generalized Ebeling formula for Coxeter polynomials

Established decompositions into ramified continued fractions

Connected Poincare series with Milnor invariants and singularity theory

Abstract

For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about the factorization of the Alexander-Conway polynomial permit to connect Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. One uses reconstractions of A'Campo links and Coxeter links. There is formulated a conjecture connecting obtained formulae with Poincare series of ring of functions on singularities through results of S. M. Gusein-Zade, F. Delgado, and A. Campillo.