Crosscap number

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In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of

1 − χ(S),

taken over all compact, connected, nonorientable surfaces S bounding K; here χ is the Euler characteristic. The crosscap number of the unknot is defined to be zero.

[edit] 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

[edit] References

• 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)