Removing even crossings, continued

In this paper we investigate how certain results related to the Hanani-Tutte theorem can be lifted to orientable surfaces of higher genus. We give a new simple, geometric proof that the weak Hanani-Tutte theorem is true for higher-genus surfaces. We extend the proof to prove that bipartite generalized thrackles in a surface S can be embedded in S. We also show that a result of Pach and Tóth that allows the redrawing of a graph removing intersections on even edges remains true on highergenus surfaces. As a consequence, we can conclude that crS(G), the crossing number of the graph G on surface S, is bounded by 2 ocrS(G) 2, where ocr(G)S is the odd crossing number of G on surface S. Finally, we begin an investigation of optimal crossing configurations for which ocr ∼ = cr