\input zb-basic \input zb-ioport \iteman{io-port 05763167} \itemau{Pelsmajer, Michael J.; Schaefer, Marcus; \v{s}tefankovi\v{c}, Daniel} \itemti{Removing even crossings.} \itemso{Felsner, Stefan (ed.), 2005 European conference on combinatorics, graph theory and applications (EuroComb '05). Extended abstracts from the conference, Technische Universit\"at Berlin, Berlin, Germany, September 5--9, 2005. Paris: Maison de l'Informatique et des Math\'ematiques Discr\`etes (MIMD). Discrete Mathematics \& Theoretical Computer Science. Proceedings. AE, 105-110, electronic only (2005).} \itemab Summary: An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and T\'oth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte. \itemrv{~} \itemcc{} \itemut{Hanani's theorem; Tutte's theorem; even crossings; crossing number; odd crossing number; independent odd crossing number} \itemli{http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAE0121} \end