the stub at a bivalent node is inverted. In particular, the arrow on the null edge in Fig. \[vestigialstub\] can be inverted without changing the value.

Kets to bras {#ketstobras}
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The intertwiner $K:\CS_j\otimes\CS_j\to\Complexes$ has an alternative interpretation as a map $K_1:\CS_j\to\CS^*_j$, where for simplicity we have dropped the prime on the second factor, and where the $1$-subscript distinguishes the new map from the old. This alternative interpretation is natural in spin network language, as seen in Fig. \[ket2bra\]. The spin network for $\ket{\psi} \in \CS_j$ is plugged into the second operand of the spin network for $K$, resulting in a spin network with one free bra chevron. The choice of the second operand of $K$ for this purpose is conventional. The result is an element of $\CS_j^*$, that is, it is a bra. As indicated in the figure, we abbreviate this spin network by drawing the same spin network for $\ket{\psi}$ that we started with, except the ket chevron is converted into a bra chevron.

In other words, when we use $K_1$ to convert a ket into a bra, we just flip the ket chevron, leaving everything else the same. In particular, we do not put a star on 