BMW closure trace and stabilization scalars

By [F1], the tangle class ${\mathrm{\Phi }}_n(x)$ depends only on the BMW algebra element $x$, so closing it and applying $P$ defines a $T$-linear scalar. For $x,y\in BM{W}_n$, the two closed tangles obtained by closing the stacked products ${\mathrm{\Phi }}_n(x){\mathrm{\Phi }}_n(y)$ and ${\mathrm{\Phi }}_n(y){\mathrm{\Phi }}_n(x)$ differ only by moving the chosen cut between the factors once around the closed tangle. This is an isotopy of the closed diagram and preserves its framing; therefore $P(\widehat{{\mathrm{\Phi }}_n(xy)})=P(\widehat{{\mathrm{\Phi }}_n(yx)})$, proving cyclicity after multiplication by $d^{-n}$.

Under tower inclusion, the new straight strand closes to a disjoint trivial circle. By [F2], $P(\widehat{{\mathrm{\Phi }}_{n+1}({\iota }_n(x))})=dP(\widehat{{\mathrm{\Phi }}_n(x)})$. Thus $d^{-(n+1)}dP(\widehat{{\mathrm{\Phi }}_n(x)})={\tau }_n(x)$, which proves tower compatibility.

Closing ${\iota }_n(x)g_n$ turns the added positive crossing into a positive curl on the last closure arc of $\widehat{{\mathrm{\Phi }}_n(x)}$; closing ${\iota }_n(x)g_n^{-1}$ gives a negative curl on the same arc. By [F2] and [F3], the respective scalar factors are $a$ and $a^{-1}$. Normalization by $d^{-(n+1)}$ gives $a{d}^{-1}{\tau }_n(x)$ and $a^{-1}{d}^{-1}{\tau }_n(x)$.

In the closure of ${\iota }_n(x)e_n$, the added cup and cap and the two added closure arcs form a detour joining the top and bottom ends of the last strand of ${\mathrm{\Phi }}_n(x)$. Following that component through the cup, cap, and closure arcs gives exactly the same connection as the untwisted closure arc for ${\mathrm{\Phi }}_n(x)$; the other components and crossings are unchanged. Pulling this detour into the closing collar is a planar isotopy, so the unnormalized evaluations agree. The normalization changes from $d^{-n}$ to $d^{-(n+1)}$, giving $d^{-1}{\tau }_n(x)$.