Average crossing number
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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
where dA is the area form on the
Alternative formulation
A less intuitive but computationally useful definition is an integral similar to the
A derivation analogous to the derivation of the linking integral will be given. Let K be a knot, parameterized by
Then define the map from the torus to the 2-sphere
by
(Technically, one needs to avoid the diagonal: points where s = t.) We want to count the number of times a point (direction) is covered by g. This will count, for a generic direction, the number of crossings in a knot diagram given by projecting along that direction. Using the
References
Further reading
- Buck, Gregory; Simon, Jonathan (1999), "Thickness and crossing number of knots", Topology and Its Applications, 91 (3): 245–257, MR 1666650.
- Ernst, C.; Por, A. (2012), "Average crossing number, total curvature and ropelength of thick knots", Journal of Knot Theory and Its Ramifications, 21 (3): 1250028, 9, MR 2887660.
- Diao, Yuanan; Ernst, Claus (2001). "The Crossing Numbers of Thick Knots and Links". In Jorgr Alberto Calvo; Kennrth C. Millet; Eric J. Rawdon (eds.). Physical Knots: Knotting, Linking, and Folding Geometric Objects in R3. Contemporary Mathematics. Vol. 304. Las Vegas, Nevada. ISBN 0-8218-3200-X.).
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: CS1 maint: location missing publisher (link - O’Hara, Jun (2003). Energy of knots and conformal geometry. K&E Series on Knots and Everything. Vol. 33. Singapore: World Scientific Publixhing Co. Pte. Ltd. ISBN 981-238-316-6..