four Reidemeister moves.

In order to show that $\widehat{L}_{\varepsilon}(f)$ is invariant under the fifth move, we must show that the value is unchanged when any edge of $f(K_7)$ is pulled over or under a given vertex $v$. An example is illustrated in Figure \[FigureK7\_loopover\].

![$K_7$ with an edge pulled over a vertex.[]{data-label="FigureK7_loopover"}](K7_loopover.eps){width="55.00000%"}

Pulling a given edge $e$ over a vertex will generate six new crossings. In Figure \[FigureK7\_loopover\] the edge $x_4$ has new crossings with the edges $x_1$, $y_1$, $z_1$, $x_2$, $y_3$, and $z_4$. The crossings with edges pointed away from the vertex $v$ ($x_1$, $y_1$, $z_1$) will have an opposite sign compared to the crossings with edges pointed toward the vertex $v$ ($x_2$, $y_3$, $z_4$). Thus, the overall change in $\widehat{L}_{\varepsilon}(f)$ is found by adding the epsilon coefficients for the crossings of $x_4$ with $x_1$, $y_1$, and $z_1$ and subtracting the epsilon coefficients for the crossings of $x_4$ with $x_2$, $y_3$, and $z_4$. It is easy to check that $\widehat{L}_{\varepsilon}(f)$ is unchanged in each case.

It follows from Lemma \[k7 invariant\], that $\widehat{L}_{\varepsilon}(f)$ is a generalized Simon invariant.

[ One can check that the epsilon coefficients we have given for $K_7$ define a homomorphism from the free $\mathbb{Z}$-module $L(K_7)$ 