Turaev surface
= Turaev surface
{c}
{title2=$2-2g_T(D)=s_A+s_B-c(D)$}
Replace each crossing of a diagram by a saddle joining its A smoothing above the projection <sphere> to its B smoothing below; cap the upper and lower state <circles>. For a connected classical <link diagram> this constructs a connected closed orientable <surface>. Its Euler characteristic is $s_A+s_B-c(D)$, so $s_A+s_B\leq c(D)+2$. For an alternating diagram the <surface> has genus zero.