Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 14 / 5 / 3 / Solution

The dual Thurston polytope is the polar of the Thurston norm unit ball. More intrinsically, in the real dual of $H_2(Y,\partial Y)$ it isThe pairing can be regarded as evaluation of $H^2(Y,\partial Y;\mathbb{R})$ on relative homology. Its definition remains valid when the Thurston norm has a kernel: the polytope then lies in the annihilator of that kernel.

Identify $Y$ with the pair of pants product $P\times S^1$. A regular Seifert fiber has homology class $h={\mu }_1+{\mu }_2+{\mu }_3$. Let $e_i$ be the relative homology class corresponding by Poincare-Lefschetz duality to the homomorphism taking the $i$th meridian to one and the other two to zero. A spanning disk for $L_1$ punctured once by each of $L_2,L_3$ is an embedded pair of pants $F$ representing $e_1$, with $c(F)=1$.

Take two arcs in $P$, one joining boundary one to boundary two, the other joining boundary one to boundary three. Their products with $S^1$ are embedded vertical surfaces in a Seifert fibered space, namely annuli $A_{12},A_{13}$. Orient them so that their relative classes are $e_1-e_2$ and $e_1-e_3$. They cost zero. Thus the three required inequalities, with ${\alpha }_i=\alpha (e_i)$, areFor completeness they give the whole polytope. Oriented cut-and-paste of copies of $F,A_{12},A_{13}$ gives the upper bound $\Vert a{e}_1+b{e}_2+c{e}_3{\Vert }_T\le |a+b+c|$. For the reverse bound, compress a minimizing surface and use the classification of incompressible surfaces in Seifert fibered spaces. Its horizontal components cover $P$ and have negative Euler characteristic equal to their unsigned covering degree; its vertical components have zero cost and zero intersection with a regular Seifert fiber. The total signed horizontal degree is $a+b+c$, so its cost is at least $|a+b+c|$. HenceIt is a line segment, because this Thurston norm has a two-dimensional kernel.