= Kirby color handle-slide identity
{c}
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 $p_ci_c=1_c$ and $\sum_ci_cp_c=1_{b\otimes a}$. 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 $V_c^bU_c^b=(d_c/d_b)1_b$. Completeness of these dual fusion maps therefore gives $\sum_b d_bU_c^bV_c^b=d_c1_{c\otimes a}$. 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.
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