All links are semiholomorphic

TL;DR

This paper proves that every link in the 3-sphere can be realized as the link of a weakly isolated singularity of a semiholomorphic polynomial, extending classical results to a broader class of functions.

Contribution

It establishes the semiholomorphic analogue of a fundamental theorem, showing all links are algebraic in this new setting, with a constructive proof providing degree bounds.

Findings

Every link type in the 3-sphere arises from a semiholomorphic polynomial.

Constructive method yields explicit polynomials for given links.

Provides an upper bound on polynomial degree for the constructed functions.

Abstract

Semiholomorphic polynomials are functions $f:\mathbb{C}^2\to\mathbb{C}$ that can be written as polynomials in complex variables $u$, $v$ and the complex conjugate $\overline{v}$. We prove the semiholomorphic analogoue of Akbulut's and King's "All knots are algebraic", that is, every link type in the 3-sphere arises as the link of a weakly isolated singularity of a semiholomorphic polynomial. Our proof is constructive, which allows us to obtain an upper bound on the polynomial degree of the constructed functions.