Easy lemma for trivalent graphs in colored Jones polynomial

Question

In his 2008 paper,

Tanaka, Toshifumi, The colored Jones polynomials of doubles of knots, J. Knot Theory Ramifications 17, No. 8, 925-937 (2008). ZBL1149.57023.

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.

Answer 1

Rewrite each edge of the graph, labeled by $k$, as $k$ strands with the $k$-th JW idempotent in the middle. Make a similar modification at the vertices. Expand the sums appearing to one side of the $2n$ strand. Each of the resulting diagrams will have a strand which leaves the $2n$ idempotent at position $i$ and returns at position $i+1$. A basic property of JW idempotents implies that this summand corresponding to this diagram is zero.

If the above does not make sense to you, then the book by Kauffman and Lins is a good reference for basic skein theory and arguments of this type.