y_7}$ as the [*1-star*]{} since these edges skip over one vertex relative to the cycle $\overline{x_1x_2...x_7}$. Similarly, we refer to the Hamiltonian cycle $\overline{z_1 z_2 ... z_7}$ as the [*2-star*]{} since these edges skip over two vertices relative to the cycle $\overline{x_1x_2...x_7}$. For consistency, we also use the term [*0-star*]{} to refer to the Hamiltonian cycle $\overline{x_1x_2...x_7}$. We orient the edges around each of the stars as illustrated. Note that this classification of oriented edges is only dependent on our initial choice of an oriented 0-star.

![An illustration of the oriented $K_7$, with the 0-star in black, the 1-star in bold, and the 2-star in grey.[]{data-label="FigureK7_labeled"}](labeledK7.eps){width="35.00000%"}

We define the [*epsilon coefficient*]{} of a pair of disjoint edges by the function:

$$\varepsilon(x_i,x_j) = \varepsilon(y_i,y_j) = \varepsilon(z_i,z_j) = \varepsilon(x_i,z_j) = \varepsilon(y_i,z_j) = 1$$ $$\varepsilon(x_i,y_j) = -1.$$

Given an oriented 0-star and an embedding $f: K_7 \rightarrow {S}^3$ with a regular projection, we define the integer $\widehat{L}_{\varepsilon}(f)$ by

$$\widehat{L}_{\varepsilon}(f) = \sum_{a\cap b=\emptyset} \varepsilon(a,b) \ell(f(a),f(b)).$$

\[k7 invariant\] Consider $K_7$ with a fixed choice of an oriented $0$-star. Then for any embedding $f:K_7\to S^3$, the value of $\widehat{L}_{\varepsilon}(f)$ is an ambient isotopy invariant.

It is easy to check that $\widehat{L}_{\varepsilon}(f)$ is invariant under the first 