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average crossing number
In the mathematical subject of knot theory, the average crossing number of a knot is the result of averaging over all directions the number of crossings in a knot diagram of the knot obtained by projection onto the plane orthogonal to the direction. The average crossing number is often seen in the context of physical knot theory.
==Definition==
More precisely, if ''K'' is a smooth knot, then for almost every unit vector ''v'' giving the direction, orthogonal projection onto the plane perpendicular to ''v'' gives a knot diagram, and we can compute the crossing number, denoted ''n''(''v''). The average crossing number is then defined as the integral over the unit sphere:
:
where ''dA'' is the area form on the 2-sphere. The integral makes sense because the set of directions where projection doesn't give a knot diagram is a set of measure zero and ''n''(''v'') is locally constant when defined.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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