Complex logarithm: Difference between revisions

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In the [[mathematics|mathematical]] field of [[knot theory]], the '''crosscap number''' of a [[knot (mathematics)|knot]] ''K'' is the minimum of
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:<math>1 - \chi(S), \, </math>
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taken over all [[compact space|compact]], [[connected space|connected]], [[orientability|non-orientable]] [[surface]]s ''S'' bounding ''K''; here <math>\chi</math> is the [[Euler characteristic]]. The crosscap number of the [[unknot]] is zero, as the Euler characteristic of the disk is one.
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==Examples==
*The crosscap number of the [[trefoil knot]] is 1, as it bounds a [[Möbius strip]] and is not trivial.
*The crosscap number of a [[torus knot]] was determined by M. Teragaito.
 
The formula for the [[knot sum]] is
:<math>C(k_1)+C(k_2)-1 \leq C(k_1 \# k_2) \leq C(k_1)+C(k_2). \, </math>
 
==Further reading==
*Clark, B.E. "Crosscaps and Knots", Int. J. Math and Math. Sci, Vol 1, 1978, pp 113&ndash;124
*Murakami, Hitoshi and Yasuhara, Akira. "Crosscap number of a knot," Pacific J. Math. 171 (1995), no. 1, 261&ndash;273.
*Teragaito, Masakazu. "Crosscap numbers of torus knots," Topology Appl. 138 (2004), no. 1&ndash;3, 219&ndash;238.
*Teragaito, Masakazu and Hirasawa, Mikami. "Crosscap numbers of 2-bridge knots," Arxiv:math.GT/0504446.
*J.Uhing. [http://jason-uhing.de/ "Zur Kreuzhaubenzahl von Knoten"], diploma thesis, 1997, University of Dortmund, (German language)
 
==External links==
*"[http://www.indiana.edu/~knotinfo/descriptions/crosscap_number.html Crosscap Number]", ''KnotInfo''.
 
{{Knot theory}}
[[Category:Knot invariants]]
 
{{knottheory-stub}}

Revision as of 08:47, 23 February 2014

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