---
tag: 9B4A2WHAPQ
kind: concept
name: Thurston norm
msc: [57K30, 57K31]
sources:
  - url: https://en.wikipedia.org/wiki/Thurston_norm
same_as:
  wikidata: Q28401506
  wikipedia: Thurston norm
---

## Summary

The Thurston norm is a [[seminorm|FP5GVCN8GV]] on the second homology of a compact orientable [[3-manifold|R20TB2KEQQ]].
It measures the least complexity, in terms of [[Euler characteristic|RNDDWDAFJ5]], of an embedded surface representing a given class.
Thurston showed that its unit ball is a polyhedron.
The classes represented by fibers of fibrations over the circle fill the cones over certain top-dimensional faces.
It connects the topology of surfaces in $3$-manifolds with fibrations over the circle.

## Explanations

### L1 human

Every $2$-dimensional homology class of a $3$-manifold is represented by embedded surfaces, often in many ways.
The Thurston norm of the class is the smallest complexity of such a surface, where a surface of genus $g \ge 1$ counts $2g - 2$ and spheres and disks count nothing.
For the complement of a nontrivial knot, the norm of the generator is $2g - 1$ with $g$ the genus of the knot, so the norm detects the simplest Seifert surface.