Crosscap number

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.


The formula for the knot sum is


Further readingEdit

  • 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 linksEdit