In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of
taken over all compact, connected, nonorientable 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.
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Concordance Crosscap Number of a Knot (part 1/3)

AlgTop21: The twoholed torus and 3crosscaps surface

AlgTop21: The twoholed torus and 3crosscaps surface
Transcription
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 2bridge knots," Arxiv:math.GT/0504446.
 J.Uhing. "Zur Kreuzhaubenzahl von Knoten", diploma thesis, 1997, University of Dortmund, (German language)
External links
 "Crosscap Number", KnotInfo.