= Thurston norm of a pair-of-pants product
{c}
{title2=$\|u\|_T=|u\cdot h|$}
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 $\|u\|_T=|u\cdot h|$. The <dual Thurston polytope> is therefore the segment between the two functionals $\pm h$ under the natural pairing.
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