Thurston norm of a pair-of-pants product ($\Vert u{\Vert }_T=|u\cdot h|$)

For $Y=P\times S^1$ with $P$ a pair of pants and $h$ its circle fiber, a horizontal surface in a Seifert fibered space of degree $d$ has ${\chi }_{-}=|d|$; vertical surfaces in a Seifert fibered space have zero cost. Cutting and pasting these representatives, and classifying essential surfaces, proves $\Vert u{\Vert }_T=|u\cdot h|$. The dual Thurston polytope is therefore the segment between the two functionals $\pm h$ under the natural pairing.