Warp-like pattern in a closed curve Given a closed curve in 2D space that intersects itself (transversally, and there's no point in which three paths or more meet), is it possible to look at it as a Celtic knot so when you follow it, one time you're above the other path in an intersection point, and one time you're under it? My gut feeling tells me it is possible, but I could not proof it. Any idea? For example this closed curve has been drawn so the crossings alternate over-under-over-under: ![](

A (reasonably well-behaved) self-intersecting closed curve divides the plane into a number of components that can be coloured alternately black and white in a checkerboard manner. Declare a segment of the curve to go over or under a crossing depending on whether black is to the left or the right of the segment as it approaches the crossing. This produces a knot that is alternating by construction.

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