the [*Simon invariant*]{}.

![The Reidemeister moves for embedded graphs.[]{data-label="Figuremy5radmoves"}](my5radmoves.eps){width="60.00000%"}

By using Simon’s method we can create similar invariants for many other embedded graphs. In particular, let $G$ be an oriented graph and let $f$ be an embedding of $G$ in $S^3$. If we can define a function $\varepsilon(a,b)$ from the set of pairs of disjoint edges of $G$ to the integers such that for any projection of $f(G)$ the value of

$$\widehat{L}_{\varepsilon}(f) = \sum_{a\cap b=\emptyset} \varepsilon(a,b) \ell(f(a),f(b))$$ is invariant under the five Reidemeister moves, then we say that $\widehat{L}_{\varepsilon}(f)$ is a [*generalized Simon invariant of $f(G)$*]{}. If for every embedding $f$ of $G$, $\widehat{L}_{\varepsilon}(f)$ is a [*generalized Simon invariant of $f(G)$*]{}, then we say that $\widehat{L}_{\varepsilon}(f)$ is a [*generalized Simon invariant of $G$.*]{}

Observe that the reduced Wu invariants given in Example \[K5K33\_2\] are identical to their Simon invariants. In fact, every reduced Wu invariant with respect to a given homomorphism $\varepsilon$ is a generalized Simon invariant with epsilon coefficients given by $\varepsilon(a,b)$. However, not every generalized Simon invariant is necessarily a reduced Wu invariant. In order to distinguish these two types of invariants, we use $\tilde{\mathcal L}_{\varepsilon}(f)$ to denote a reduced Wu invariant and $\widehat{L}_{\varepsilon}(f)$ to denote a generalized 