In this paper we investigate how certain results related to the HananiTutte 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...

The theory of zero-crossing detection is extended to include the first and second spatial derivative of the gaussian distribution. On the premise that phase contains a considerable proportion of the information content in an unknown signal, we show that phase can be extracted from the first and second derivatives of an appropriate filter. It is shown that the spatial gradient of phase can be us...

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 Tóth 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 the...

Let K′ be a knot that admits no cosmetic crossing changes and let C be a prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K′ admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-cable knots.