The dual 1-form of a path

To each path γ=(e1,,en) we can associate a dual 1-form γ^ given by

γ^(e)=#{k{1,,n}|e=ek}#{k{1,,n}|e=ρ(ek)}.

Here are two examples: The following path γ passes through one edge e2=e9 twice:

The corresponding dual 1-form γ^ has evaluates to 2 on this edge.

The second example contains the edges e2 and e8=ρ(e2).

In the corresponding dual 1-form these two edges cancel:

Let us calculate the derivative of the dual 1-form γ^ associated to a path γ:

dγ^(v)=evγ^(e)=ev(#{k{1,,n}|e=ek}#{k{1,,n}|e=ρ(ek)})=#{k{1,,n}|end(ek)=v}#{k{1,,n}|start(ek)=v}.

If γ is closed, this will be zero for all vertices v. This is because a closed path enters any vertex as often  as it leaves it. On the other hand, if γ is not closed, the dγ^ evaluates to 1 on the vertex end(γ) to 1 on start(γ) and to zero on all other vertices.

In particular this implies that for a closed path γ the dual 1-form γ^ is also closed, i.e. γ^=0.

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