generality, we may assume that an edge $e$ is pulled over a vertex $v$ and the adjacent outer edges point towards $v$ (see Figure \[FigureM\_2N+1\_loopover\]). Pulling $e$ over $v$ generates three new crossings: two with outer edges and one with an inner edge. We must determine the change in $\widehat{L}_{\varepsilon}(f)$ as a result of of these added crossings.

![$M_{2N+1}$ with an edge $e$ pulled over a vertex $v$.[]{data-label="FigureM_2N+1_loopover"}](M14_oriented_loopover.eps){width="30.00000%"}

Below we compute the possibilities for this change $\Delta \widehat{L}_{\varepsilon}(f)$, and show that in all cases this value is zero. The crossings between the edge $e$ and the two outer edges have the same sign while the crossing of $e$ with an inner edge has the opposite sign. The epsilon coefficients for the crossings of $e$ with the two outer edges are given in parenthesis (with the edge whose minimal outer edge distance from $e$ is larger given first, and the edge closer to $e$ given second), while the epsilon coefficient of the crossing of $e$ with the inner edge is given afterward. For ease of notation, let $d(e,v)$ denote the minimum number of edges in any path between $e$ and $v$ using only outer edges and not counting $e$.

-   $e$ 