is negative.}
\end{cases}$$ Here (and in what follows), a monoidal product indexed by a fully ordered set is taken following the order of the set of indices (as in Section \[sect-conventions\]). Note that the naturality of symmetry and the definition of $\sigma$ imply that $$P_\sigma S_{\mathcal{L}}= S_{\mathcal{U}}P_\sigma.$$ For $l \in {\mathcal{L}}$ and $u \in {\mathcal{U}}$, consider the integers $|u|$ and $|l|$ defined in Section \[sect-Heegaard-diag\] and the morphisms $\Delta_{|l|}$ and $\mu_{|u|}$ defined in Section \[sect-Hopf-alg\]. Recall that $N$ is computed by $$N=\sum_{u \in {\mathcal{U}}} |u|=\sum_{l \in {\mathcal{L}}} |l|.$$ Set $$\Delta_{\mathcal{L}}=\bigotimes_{l \in {\mathcal{L}}} \Delta_{|l|}  {\colon}A^{\otimes g} \to A^{\otimes N}  {\quad \text{and} \quad}\mu_{\hspace{1pt}{\mathcal{U}}}=\bigotimes_{u \in {\mathcal{U}}} \mu_{|u|} {\colon}A^{\otimes N} \to A^{\otimes g}.$$ Finally, define $$\label{eq-KAD}
K_A(D)=\mu_{\hspace{1pt}{\mathcal{U}}} P_\sigma S_{\mathcal{L}}\Delta_{\mathcal{L}}=  \mu_{\hspace{1pt}{\mathcal{U}}} S_{\mathcal{U}}P_\sigma  \Delta_{\mathcal{L}}{\colon}A^{\otimes g} \to A^{\otimes g}.$$

Examples {#sect-ex-KAD}
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1\) Let ${\mathbb{T}}={\mathbb{R}}^2/{\mathbb{Z}}^2$ be the torus with orientation induced by the counterclockwise orientation of ${\mathbb{R}}^2$. Consider the following circles on ${\mathbb{T}}$: $$u= (1/2,0)+{\mathbb{R}}(0,1) + {\mathbb{Z}}^2 {\quad \text{and} \quad}l={\mathbb{R}}(0,1) + {\mathbb{Z}}^2.$$ Then $D_0=({\mathbb{T}},\{u\},\{l\})$ is a Heegaard diagram which represents $S^1 \times S^2$. Since $D_0$ has no intersection points, we obtain that $$K_A(D_0)=\mu_0 \,\Delta_0= \;{\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{KAD-S1S2.eps}}}}$$ for any choice for the orientations and base points of the circles.

2\) Consider as above the oriented torus ${\mathbb{T}}={\mathbb{R}}^2/{\mathbb{Z}}^2$. Pick a 