= Kauffman bracket breadth bound
{c}
{title2=$\operatorname{br}_A\langle D\rangle\leq2c(D)+2s_A+2s_B-4$}
A state with $k$ B smoothings has at most $s_A+k$ <circles>, so its largest A exponent is at most $c(D)+2s_A-2$. The dual argument bounds the smallest exponent below by $-c(D)-2s_B+2$. For a connected <link diagram>, the <Turaev surface> inequality bounds the resulting breadth by $4c(D)$. For a connected <reduced alternating diagram>, adequacy prevents cancellation at both extremes and planar <Euler characteristic> gives equality. A disconnected projection needs the additional term in the <Jones polynomial breadth bound for a disconnected diagram>.
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