In his 2008 paper,
Tanaka stated a lemma (Lemma 3.3) saying that if two planar, banded trivalent graphs are connecting by a single strand colored with $2n$, then the entire diagram has bracket 0 unless $n=0$, in which case the diagram reduces to the disjoint union of the two `summands'.
Tanaka claimed that this lemma can be easily verified by the properties of the Jones-Wenzl idempotents. However, it is not clear to me which properties he used to prove the lemma.
We've tried to use the fact that a strand leaving and entering the same Jones-Wenzl idempoent is equal to zero, but cannot manage to argue why this always occurs given a nonzero bridging strand colored with an even number. Any insight or familiarity with this basic result in graphical skein theory would be appreciated.