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On real algebraic links in the 3-sphere associated with mixed polynomials

Raimundo N. Aráujo dos Santos, Eder L. Sanchez Quiceno

TL;DR

This work tackles the problem of realizing fibered links in $S^3$ as real algebraic links via mixed polynomial germs with isolated singularities. It develops a robust framework based on the Newton boundary, non-degeneracy (including strong non-degeneracy), and radial/polar weights, and proves two main criteria (a semiholomorphic Criterion and an axis-isolation Criterion) that guarantee strong realizations for broad classes of mixed polynomials. Central to the results is the analysis of products $f=f_p\cdot q$, where careful coordination of the weight sequences and face functions yields $\mathcal{A}(f)=N^+$ and hence strong realizations of the associated links, with explicit constructions drawn from braids and radial semiholomorphic factors. The paper thereby extends known families of real algebraic links, offers concrete methods for building strong realizations, and provides evidence supporting the Benedetti–Shiota conjecture by encompassing new classes of fibered links within real algebraic realizations.

Abstract

In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the $3$-sphere. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti-Shiota conjecture, which state that any fibered link on the $3$-sphere is a real algebraic link.

On real algebraic links in the 3-sphere associated with mixed polynomials

TL;DR

This work tackles the problem of realizing fibered links in as real algebraic links via mixed polynomial germs with isolated singularities. It develops a robust framework based on the Newton boundary, non-degeneracy (including strong non-degeneracy), and radial/polar weights, and proves two main criteria (a semiholomorphic Criterion and an axis-isolation Criterion) that guarantee strong realizations for broad classes of mixed polynomials. Central to the results is the analysis of products , where careful coordination of the weight sequences and face functions yields and hence strong realizations of the associated links, with explicit constructions drawn from braids and radial semiholomorphic factors. The paper thereby extends known families of real algebraic links, offers concrete methods for building strong realizations, and provides evidence supporting the Benedetti–Shiota conjecture by encompassing new classes of fibered links within real algebraic realizations.

Abstract

In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the -sphere. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti-Shiota conjecture, which state that any fibered link on the -sphere is a real algebraic link.

Paper Structure

This paper contains 8 sections, 14 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mixed Polynomial and Newton Boundary
  3. Newton boundary and non-degeneracy of mixed polynomials.
  4. Newton boundary for products of mixed polynomials
  5. Mixed Isolated Singularity
  6. Mixed singularities outside of the axes
  7. Mixed singularities at the axes
  8. Real Algebraic Links

Key Result

Theorem 1.2

Let $f:(\mathbb{C}^2,0) \to (\mathbb{C},0)$ be a germ of mixed polynomials such that $\mathcal{N}(f) \subset \mathcal{A}(f)$. Then $\mathcal{A}(f)=N^+$. In particular, if $\mathcal{N}(f) \subset \mathcal{P}(f)$ then $\mathcal{A}(f)=N^+$.

Figures (4)

  • Figure 1: Newton boundary of $f_u$ and $r_{f}$.
  • Figure 2: Newton boundary of $f$.
  • Figure 3: Square braid formed by the Fourier parametrization of $(F_B(\frac{t+2\pi j}{3}),G_B(\frac{t+2\pi j}{3}),t) \ j=1,2,3$.
  • Figure 4: Braid representation of $L_{f_{k,5,5}}$ with $(u,t) \in \mathbb{C} \times [0,2\pi]$ when $v=r_{0} e^{it}, \text{ and } r_0 \ll 1$.

Theorems & Definitions (37)

  • Conjecture 1.1: Benedetti_Shiota1998, p. 588
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2: Oka2010
  • Definition 2.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 27 more