Crosscap number

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

$1 - \chi(S), \,$

taken over all compact, connected, non-orientable surfaces S bounding K; here $\chi$ 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

$C(k_1)%2BC(k_2)-1 \leq C(k_1 \# k_2) \leq C(k_1)%2BC(k_2). \,$

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.

Further reading

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)