Thurston norm ($\Vert \cdot {\Vert }_T$)

For a compact oriented three-manifold, the integral relative class $u$ has norm the minimum of ${\chi }_{-}(F)={\sum }_j\max (0,-\chi (F_j))$ over properly embedded oriented topological surfaces representing $u$. Homogeneity and continuity extend it to real relative homology. It is a seminorm, since spheres, disks, annuli and tori have zero cost.

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.

In an irreducible three-manifold with incompressible boundary, a minimizing surface can be cut along a separating incompressible torus after removing inessential intersection circles. The essential pieces have no disk or sphere components, so their ${\chi }_{-}$ costs add. Minimizing independently in the pieces and gluing their matching boundary curves gives the corresponding upper bound. This allows Thurston norm calculations for satellite knots.

The dual polytope consists of the real linear functionals $\alpha$ on $H_2(Y,\partial Y;\mathbb{R})$ such that $|\alpha (u)|\le \Vert u{\Vert }_T$ for every $u$. Thus it is the polar of the unit ball of the Thurston norm and annihilates that seminorm's kernel.