Geometric intersection number ( $i (α, β)$ )

The geometric intersection number

i (α, β)

is the minimum number of transverse intersection points among representatives of the isotopy classes of

α

and

β

. Representatives in minimal position realize it.

Table of contents
- Algebraic intersection number of curves on an oriented surface Geometric intersection number
  - Isotopy invariance of algebraic intersection number Algebraic intersection number of curves on an oriented surface

Algebraic intersection number of curves on an oriented surface ( $⟨ α, β ⟩$ )

For transverse oriented curves on an oriented surface, the algebraic intersection number is the sum of the local signs of their crossings. It depends only on their oriented homology classes and is antisymmetric:

⟨ α, β ⟩ = - ⟨ β, α ⟩ .

Its absolute value is at most the geometric intersection number.

Isotopy invariance of algebraic intersection number

During a generic isotopy of one transverse curve, crossings with a fixed curve can only be created or removed in pairs of opposite local sign. Their signed sum, and hence the algebraic intersection number, is therefore invariant.

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