Turaev surface ($2-2g_T(D)=s_A+s_B-c(D)$)

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 $s_A+s_B\le c(D)+2$. For an alternating diagram the surface has genus zero.