A geometric characterization of the upper bound for the span of the Jones polynomial

Understand

Let D be a link diagram with n crossings, s_A and s_B its extreme states and |s_AD| (resp.

• |s_BD|) the number of simple closed curves that appear when smoothing D according to s_A (resp.
• s_B).
• We give a general formula for the sum |s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary.