Topology of strongly polar weighted homogeneous links

TL;DR

This paper studies the topology of strongly polar weighted homogeneous links arising from a specific $S^1$ action on $S^3$, analyzing their structure, degenerations, and relation to mixed polynomial links.

Contribution

It introduces a detailed topological analysis of strongly polar weighted homogeneous links and explores their degeneration relations within the context of $S^1$ actions.

Findings

Characterization of the space of such links

Identification of smooth degeneration relations

Connection to mixed polynomial links

Abstract

We consider a canonical $S^1$ action on $S^3$ which is defined by $(\rho,(z_1,z_2))\mapsto (z_1\rho^p,z_2\rho^q)$ for $\rho\in S^1$ and $(z_1,z_2)\in S^3\subset {\mathbb C}^2$. We consider a link consisting of finite orbits of this action, which some of the orbits are reversely oriented. Such a link appears as a link of a certain type of mixed polynomials. We study the space of such links and show smooth degeneration relations.