Alternating links on nonorientable surfaces and Klein-bottly alternating links

It has been known for several decades that classical alternating links in the 3-sphere have nice hyperbolic geometric properties. Recent work generalises such results to give hyperbolic geometry of links with alternating projections onto any surface in very general 3-manifolds. However, the most general results require an orientable projection surface. In this paper, we extend to alternating links on nonorientable projection surfaces. As an application, we study Klein-bottly alternating links in prism manifolds, which are a natural generalisation of Adams’ toroidally alternating links in lens spaces.