Jones-Wenzl twist eigenvalue (${\theta }_n=(-1{)}^n{A}^{n(n+2)}$)

A positive framing twist on an n-strand Jones-Wenzl cable has n positive individual curls and $n(n-1)$ braid crossings. Each curl contributes $-A^3$, and each braid crossing contributes $A$ because its turnback term annihilates the Jones-Wenzl idempotent. Their product is the displayed scalar. This is the colored framing factor used in surgery invariants.