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Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is represented by a surface, and $||x||$ is declared to be minimum possible value of the negative the sum of the Euler characteristics of the components of a representing surface, throwing away any components that have positive Euler characteristic (i.e. spheres and disks).

Thurston proved that there is a finite set $B_T \subset H_1(Y; \mathbb{Z})$ such that

$$ ||x|| = \max_{\alpha \in B_T} \alpha \cdot x. $$

(Here I mean the intersection pairing).

I would like to see some examples where one can explicitly (and provably) find surfaces that are norm-minimizing and compute the Thurston polytope. I am mostly interested in cases where $Y$ is a link complement in $S^3$.

The only strategy I have now is to give an upper bound on $B_T$ by examining explicit surfaces (for link complements, I use Seifert surfaces for some subset of the components) and to give a lower bound using McMullen's theorem that the Thurston polytope contains the doubled Newton polygon of the Alexander polynomial. When the upper and lower bounds match exactly, all is well; when they don't, I have trouble making any further progress.

For concreteness, here are some possible examples that I would like to see: the complement of the $(p,q)$ torus knot, the complement of the Borromean rings, the two-component link with linking number $n$. For explicit cohomology classes, what are norm-minimizing surfaces that represent a linear combination of (the cohomology classes) of the Seifert surfaces for the knot components?

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You should look at Thurston's original paper on this topic:

  • A norm for the homology of 3-manifolds.
  • Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130.

There are multiple explicit examples just as you ask for.

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