We present a novel approach to understanding the extraordinary diversity of magnetic skyrmion solutions. Our approach combines a new classification scheme with efficient analytical and numerical methods. We introduce the concept of chiral kinks to account for regions of disfavoured chirality in spin textures, and classify two-dimensional magnetic skyrmions in terms of closed domain walls carrying such chiral kinks. In particular, we show that the topological charge of magnetic skyrmions can be expressed in terms of the constituent closed domain walls and chiral kinks. Guided by our classification scheme, we propose a method for creating hitherto unknown magnetic skyrmions which involves initial spin configurations formulated in terms of holomorphic functions and subsequent numerical energy minimization. We numerically study the stability of the resulting magnetic skyrmions for a range of external fields and anisotropy parameters, and provide quantitative estimates of the stability range for the whole variety of skyrmions with kinks. We show that the parameters limiting this range can be well described in terms of the relative energies of particular skyrmion solutions and isolated stripes with and without chiral kinks.
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