Topological and Hopf charges of a twisted Skyrmion string

We examine nonlinear sigma models, in particular the Skyrme model with a twist (the twisted Skyrmion string), which comprises a vortex solution with an added dependence on a twist term \(mkz\), where \(z\) is the vertical coordinate. The topological and Hopf charges of a twisted Skyrmion string are calculated and are shown to be equivalent to the winding number \(n\) of the vortex and proportional to \(nmk\), respectively. The principle of conservation of the Hopf charge, which is a standard property of the baby Skyrmion model, is generalised to cover the non-compact twisting solutions studied here.