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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
874:
Diao, Yuanan; Ernst, Claus (2001). "The
Crossing Numbers of Thick Knots and Links". In Jorgr Alberto Calvo; Kennrth C. Millet; Eric J. Rawdon (eds.).
519:. This will count, for a generic direction, the number of crossings in a knot diagram given by projecting along that direction. Using the
917:
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151:
764:{\displaystyle {\frac {1}{4\pi }}\int _{T^{2}}{\frac {|(f'(s)\times f'(t))\cdot (f(s)-f(t))|}{|(f(s)-f(t))|^{3}}}\,ds\,dt.}
555:. Instead of integrating this form, integrate the absolute value of it, to avoid the sign issue. The resulting integral is
329:
268:
95:
227:. The integral makes sense because the set of directions where projection doesn't give a knot diagram is a set of
36:
30:
912:. K&E Series on Knots and Everything. Vol. 33. Singapore: World Scientific Publixhing Co. Pte. Ltd.
252:
47:
937:
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Ernst, C.; Por, A. (2012), "Average crossing number, total curvature and ropelength of thick knots",
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A less intuitive but computationally useful definition is an integral similar to the
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A derivation analogous to the derivation of the linking integral will be given. Let
228:
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Buck, Gregory; Simon, Jonathan (1999), "Thickness and crossing number of knots",
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giving the direction, orthogonal projection onto the plane perpendicular to
515:.) We want to count the number of times a point (direction) is covered by
224:
138:
523:, as in the linking integral, would count the number of crossings with
876:
Physical Knots: Knotting, Linking, and
Folding Geometric Objects in R
528:
320:
878:. Contemporary Mathematics. Vol. 304. Las Vegas, Nevada.
15:
507:(Technically, one needs to avoid the diagonal: points where
497:{\displaystyle g(s,t)={\frac {f(s)-f(t)}{|f(s)-f(t)|}}.}
209:{\displaystyle {\frac {1}{4\pi }}\int _{S^{2}}n(v)\,dA}
137:). The average crossing number is then defined as the
564:
397:
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378:{\displaystyle g:S^{1}\times S^{1}\rightarrow S^{2}}
309:{\displaystyle f:S^{1}\rightarrow \mathbb {R} ^{3}.}
94:is the result of averaging over all directions the
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129:, and we can compute the crossing number, denoted
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847:Journal of Knot Theory and Its Ramifications
898:: CS1 maint: location missing publisher (
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66:Learn how and when to remove this message
114:is a smooth knot, then for almost every
29:This article includes a list of general
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910:Energy of knots and conformal geometry
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35:it lacks sufficient corresponding
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818:Topology and Its Applications
319:Then define the map from the
262:be a knot, parameterized by
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859:10.1142/S0218216511009601
223:is the area form on the
247:Alternative formulation
88:average crossing number
50:more precise citations.
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253:Gauss linking integral
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908:O’Hara, Jun (2003).
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100:physical knot theory
110:More precisely, if
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853:(3): 1250028, 9,
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229:measure zero
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127:knot diagram
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80:mathematical
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938:Knot theory
800:O’Hara 2003
143:unit sphere
116:unit vector
84:knot theory
82:subject of
48:introducing
775:References
106:Definition
31:references
894:cite book
712:−
667:−
649:⋅
626:×
582:∫
575:π
537:area form
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350:×
289:→
172:∫
165:π
141:over the
932:Category
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225:2-sphere
139:integral
125:gives a
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90:of a
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900:link
880:ISBN
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231:and
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