Links of inner non-degenerate mixed functions, Part II

TL;DR

This paper provides a complete characterization of links of singularities for inner non-degenerate mixed polynomials with nice Newton boundaries, extending previous results and introducing new concepts like P-fibered braids.

Contribution

It offers a full description of links arising from such singularities, including symmetry conditions and new braid notions, expanding the understanding of mixed polynomial singularities.

Findings

Links are characterized by symmetry conditions on sequences of associated links.

The results apply to both convenient and non-convenient mixed polynomials.

Introduction of P-fibered braids with O-multiplicities for describing semiholomorphic polynomial links.

Abstract

Let $f:\mathbb{C}^2\to\mathbb{C}$ be an inner non-degenerate mixed polynomial with a nice Newton boundary with $N$ compact 1-faces. In the first part of this series of papers we showed that $f$ has a weakly isolated singularity and that its link can be constructed from a sequence of links $L_1, L_2,\ldots,L_N$, each of which is associated with a compact 1-face of the Newton boundary of $f$. In this paper, we offer a complete description of the links of singularities of inner non-degenerate mixed polynomials with nice Newton boundary. We show that a link arises as the link of such a singularity if and only if it is the result of the procedure from Part 1 for a sequence of links $L_1,L_2,\ldots,L_N$ that all satisfy certain symmetry conditions. We prove the same result for convenient, Newton non-degenerate mixed polynomials with nice Newton boundary. We also introduce the notion of P-fibered braids with $O$-multiplicities and coefficients, which allows us to describe the links of isolated singularities of inner non-degenerate semiholomorphic polynomials (as opposed to weakly isolated singularities).