Links of inner non-degenerate mixed functions, Part II (2307.15340v2)

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).