# Crosscap number

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

${\displaystyle 1-\chi (S),\,}$

taken over all compact, connected, non-orientable surfaces S bounding K; here ${\displaystyle \chi }$ is the Euler characteristic. The crosscap number of the unknot is zero, as the Euler characteristic of the disk is one.

## Examples[]

The formula for the knot sum is

${\displaystyle C(k_{1})+C(k_{2})-1\leq C(k_{1}\#k_{2})\leq C(k_{1})+C(k_{2}).\,}$