Kirby color handle-slide identity

The bracket of a framed link with every surgery component colored by the Kirby color is unchanged by a handle slide. For a fixed self-dual color a choose inclusion/projection maps $i_c:c\to b\otimes a$, $p_c:b\otimes a\to c$ with $p_c{i}_c=1_c$ and ${\sum }_c{i}_c{p}_c=1_{b\otimes a}$. Bend the a strand to form $U_c^b:b\to c\otimes a$ and $V_c^b:c\otimes a\to b$. Closing the diagrams gives $V_c^b{U}_c^b=(d_c/d_b)1_b$. Completeness of these dual fusion maps therefore gives ${\sum }_b{d}_b{U}_c^b{V}_c^b=d_c{1}_{c\otimes a}$. Resolve the strand meeting a slide band into these channels. Moving the band across the colored loop bends its maps into U and V, and this weighted identity replaces the old loop-color sum by precisely the new Kirby coefficient $d_c$. The exterior ribbon diagrams agree by isotopy. Summing the sliding color with its own Kirby weight proves the full identity, including its carried band framing.