Surface links which are coverings over the standard torus

TL;DR

This paper introduces a novel method for constructing surface links in 4-space as branched coverings over a standard torus, revealing new properties and invariants of these links.

Contribution

It presents the concept of torus-covering links, explores their equivalences, link groups, ribbon structures, and computes quandle cocycle invariants from classical braids.

Findings

Certain torus-covering T^2-links are equivalent to split unions of spun T^2-links.

Some torus-covering T^2-links have non-classical link groups.

The paper computes quandle cocycle invariants for these links from classical braids.

Abstract

We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering $T^2$-link is equivalent to the split union of spun $T^2$-links and turned spun $T^2$-links. We show that a certain torus-covering $T^2$-link has a non-classical link group. We give a certain class of ribbon torus-covering $T^2$-links. We present the quandle cocycle invariant of a certain torus-covering $T^2$-link obtained from a classical braid, by using the quandle cocycle invariants of the closure of the braid.