the label of the ket. This distinguishes the map $K_1:\CS_j \to \CS_j^*$ from the metric or Hermitian conjugation, which is also a map $:\CS_j \to \CS_j^*$. When the metric is used to convert a ket to a bra, not only is the ket chevron flipped to a bra chevron, but a star is appended to the label. These two maps are quite distinct; in particular, $K_1$ is a linear map, while the metric is an antilinear map. As indicated in Fig. \[ket2bra\], the results are not the same.

When a ket is turned into a bra, the components with respect to some basis change from contravariant to covariant. But since there is more than one way to do this, any notation based on the position (upper or lower) of the indices is inadequate to represent the result.

The map $K_1$ can be used to turn a ket chevron into a bra chevron on any spin network, not only on kets themselves. A notation for an arbitrary spin network with an edge terminating in a ket chevron is shown in Fig. \[Xket\]. The circle around the $X$ indicates the rest of the spin network, which may include other edges terminating in 