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.

Knot sum

The crosscap number of a knot sum is bounded:

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.