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 | |||
:<math>1 - \chi(S), \, </math> | |||
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. | |||
==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 113–124 | |||
*Murakami, Hitoshi and Yasuhara, Akira. "Crosscap number of a knot," Pacific J. Math. 171 (1995), no. 1, 261–273. | |||
*Teragaito, Masakazu. "Crosscap numbers of torus knots," Topology Appl. 138 (2004), no. 1–3, 219–238. | |||
*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 19:56, 2 January 2014
In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of
taken over all compact, connected, non-orientable surfaces S bounding K; here is the Euler characteristic. The crosscap number of the unknot is zero, as the Euler characteristic of the disk is one.
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
Further reading
- Clark, B.E. "Crosscaps and Knots", Int. J. Math and Math. Sci, Vol 1, 1978, pp 113–124
- Murakami, Hitoshi and Yasuhara, Akira. "Crosscap number of a knot," Pacific J. Math. 171 (1995), no. 1, 261–273.
- Teragaito, Masakazu. "Crosscap numbers of torus knots," Topology Appl. 138 (2004), no. 1–3, 219–238.
- Teragaito, Masakazu and Hirasawa, Mikami. "Crosscap numbers of 2-bridge knots," Arxiv:math.GT/0504446.
- J.Uhing. "Zur Kreuzhaubenzahl von Knoten", diploma thesis, 1997, University of Dortmund, (German language)
External links
- "Crosscap Number", KnotInfo.