To each path $\gamma = (e_1, \ldots ,e_n)$ we can associate a dual 1-form $\hat{\gamma}$ given by $\hat{\gamma}(e) =\#\{k\in \{1,\ldots,n\} \,|\,  e=e_k\}-\#\{k\in \{1,\ldots,n\} \,|\,  e=\rho(e_k)\}$. Here are two examples: The following path $\gamma$ passes through one edge $e_2=e_9$ twice: The … Continue reading The dual 1-form of a path