Simon invariant.

We say that a graph embedded in ${S}^3$ is *achiral* if there is an orientation reversing homeomorphism of ${S}^3$ that takes the graph to itself setwise. Otherwise, we say the embedded graph is *chiral*. We say that an abstract graph is *intrinsically chiral* if every embedding of the graph in $S^3$ is chiral. Note that when we talk about chirality or achirality we are considering embedded graphs as subsets of $S^3$ disregarding any edge labels or orientations. For example, we saw in Figure \[achiral\] that $K_5$ and $K_{3,3}$ have achiral embeddings, although it was shown in [@Nikkuni; @2006] that no embedding of either of these graphs has an orientation reversing homeomorphism that preserves the edge labels and orientations given in Figure \[2K3K5K33\].

We now define generalized Simon invariants for some specific graphs and families of graphs, and use these invariants to prove that the graphs are intrinsically chiral.

The complete graph $K_7$ {#sec:K7 .unnumbered}
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Consider the complete graph $K_7$ with labeled edges as illustrated in Figure \[FigureK7\_labeled\]. We refer to the edges $x_1, x_2, ..., x_7$ as “outer edges” and the rest of the edges as “inner edges.” We refer to the Hamiltonian cycle $\overline{y_1 y_2 ... 