Braiding surface links which are coverings over the standard torus

TL;DR

This paper studies a class of surface links in 4-space called torus-covering links, presenting a method to braid them over a 2-sphere, and determines their braid index in specific cases.

Contribution

It introduces a braiding technique for torus-covering links over the 2-sphere and estimates their braid index, including exact values for certain spun torus knots.

Findings

Braid index of turned spun T^2-knot of torus (2,p)-knot is four.

Provides an upper estimate for the braid index of torus-covering links.

Includes spun and turned spun T^2-knots as special cases.

Abstract

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun $T^2$-knots and turned spun $T^2$-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an upper estimate of the braid index of a torus-covering link. In particular we show that the turned spun $T^2$-knot of the torus $(2,\,p)$-knot has the braid index four.