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The unitary matrix relating the anyon fusion trees $((ab{)}_{u}c{)}_t$ and $(a(bc{)}_v{)}_t$. Fix the convention $|((ab{)}_{u}c{)}_t⟩={\sum }_v[F_t^{abc}{]}_{uv}|(a(bc{)}_v{)}_t⟩$. An inverse change of parentheses uses the adjoint matrix, so its entries are complex conjugates with exchanged indices.

The Hilbert space of ways to combine the specified anyon charges to the specified total charge. For two inputs its dimension is the anyon fusion multiplicity $N_{ab}^c$. Repeated anyon fusion is described by an anyon fusion tree.

Read the arrows upwards through the central part of the diagram. The lower anyon fusion tree first splits $a$ into $b,c$, then $b$ into $d,j$ and $c$ into $h,e$. The upper anyon fusion tree combines $d,j$ into $f$, $h,e$ into $g$, and finally $f,g$ into $a$. The two interior crossings are not vertices. Thus admissibility requires the six anyon fusion multiplicitiesIf any factor vanishes, the diagram has zero value.

Make an anyon vertex gauge transformation by multiplying each splitting vertex $z\to x,y$ by an independent wave phase $u_z^{xy}$ and its adjoint fusion vertex by $(u_z^{xy}{)}^{-1}$. The diagram changes byTherefore the label-matching condition for cancellation of all anyon vertex gauge transformation factors iswith $a,d,e,h,j$ otherwise unrestricted apart from the anyon fusion rules. This statement also allows coincident labels: the three ordered vertex triples in the numerator must equal those in the denominator as multisets; matching the triples in any of the six possible orders still forces $b=f,c=g$. Conversely, these equalities visibly give $G=1$.

Closure of the drawing alone does not guarantee invariance under an anyon vertex gauge transformation: unmatched fusion and splitting vertices retain their independently chosen basis phases. A zero value is, of course, invariant regardless of phases; the boxed condition describes cancellation of the vertex factors themselves.

The two crossings of $j,h$ make their full anyon monodromy. For a transparent anyon $h$, this is the identity, so that pair of crossings can be removed together. No single-crossing exchange phase, and no topological spin from a twist, is introduced.

After this removal the upper and lower anyon fusion trees differ only in their intermediate labels. In the physics normalization of anyon diagrams, successive split-fuse bubbles give the Kronecker deltas ${\delta }_{bf}$ and ${\delta }_{cg}$. The remaining closed normalized state has norm one. Hence, for the admissible labelings of a,In particular, every admissible diagram satisfying the label matching in a evaluates to $1$. There is no factor $d_a$ here: the source explicitly permits the physics normalization of anyon diagrams, in which a closed normalized state has value $1$.

Use an anyon fusion tree in which the two crossing charges combine first. With external charges in the order $d,j,h,e$, define the normalized treesThe desired value is the matrix element of anyon monodromy between $|T_{bc}⟩$ and $|T_{fg}⟩$. Their closure is understood in the physics normalization of anyon diagrams.

First apply the anyon F-matrix to the three charges $d,j,c$, and then an inverse anyon F-matrix to $j,h,e$. With the source convention $|((uv{)}_{r}w{)}_t⟩={\sum }_s[F_t^{uvw}{]}_{rs}|(u(vw{)}_s{)}_t⟩$, this givesOnly allowed anyon fusion channels enter the sums: in the second tree these require $N_{jh}^x{N}_{xe}^y{N}_{dy}^a=1$. The complex conjugate in $U$ comes from the inverse change of parentheses, not from reversing a crossing.

Reading the actual overcrossings from bottom to top gives two positive exchanges in the source's anyon R-matrix convention. Their product isThe order of $h,j$ is restored after the second exchange. In the $X$ basis this anyon monodromy is diagonal, soEquivalently, entirely in terms of anyon F-matrices and anyon R-matrices,This expression is also meaningful for $b\ne f$ or $c\ne g$: it then carries the unmatched anyon vertex gauge transformation factors identified in a.

To check the transparent limit, put $m_x=1$ in every allowed channel. The unitary matrix identity for $F_y^{jhe}$ first givesOnce $c=g$, the unitary matrix identity for $F_a^{djc}$ givesas in c. The same cancellation proves that the recoupling expression is an identity operator whenever the full anyon monodromy is trivial, including when the individual exchanges have fermionic signs.

The convention in which the anyon fusion trees form an orthonormal basis and a splitting vertex followed by its adjoint fusion vertex gives the identity on the total-charge strand. A closed normalized state has norm one. It differs from the isotopy normalization of anyon diagrams, in which an unknotted closed charge-$a$ strand evaluates to $d_a$.

The identity charge of an anyon fusion rule: $1\times a=a$. In an anyon fusion tree, a strand of this charge may be erased using the unit conventions.