Then $D$ is an ordered oriented based Heegaard diagram. We have: $$\begin{aligned}
& l_1\cap (u_1 \cup u_2)=\{1<2<3<4<5<6<7\},\\
& l_2\cap (u_1 \cup u_2)=\{a<b<c<d<e\}, \\
& u_1\cap (l_1 \cup l_2)=\{1<2<3<4<a<6<c\}, \\
& u_2\cap (l_1 \cup l_2)=\{b<7<d<e<5\}.\end{aligned}$$ Thus the set of intersection points ordered by ${\mathcal{L}}$ is $$\{1<2<3<4<5<6<7<a<b<c<d<e\}$$ and the set of intersection points ordered by ${\mathcal{U}}$ is $$\{1<2<3<4<a<6<c<b<7<d<e<5\}.$$ Consequently the permutation $\sigma \in \mathfrak{S}_{12}$ associated to $D$ is the cycle $$\sigma=(5,12,11,10,7,9,8).$$ Since the positive intersection points are $1,2,3,4,d,e$ and the negative ones are $5,6,7,a,b,c$, we obtain that $$S_{\mathcal{L}}={\mathrm{id}}_{A^{\otimes 4}} \otimes S^{\otimes 6} \otimes {\mathrm{id}}_{A^{\otimes 2}}
 {\quad \text{and} \quad}S_{\mathcal{U}}={\mathrm{id}}_{A^{\otimes 4}} \otimes S^{\otimes 5} \otimes {\mathrm{id}}_{A^{\otimes 2}} \otimes S.$$ Then $$K_A(D)=(\mu_7 \otimes \mu_5) P_\sigma  S_{\mathcal{L}}(\Delta_7 \otimes \Delta_5)=(\mu_7 \otimes \mu_5) S_{\mathcal{U}}P_\sigma  (\Delta_7 \otimes \Delta_5)  {\colon}A^{\otimes 2} \to A^{\otimes 2}.$$ Graphically: $$K_A(D)=\; {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{KAD-Poincare.eps}}}} \;.$$

Good pairs and 3-manifolds invariants {#sect-inv-from-gp}
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\[sec3\]

In this section, $A=(A,\mu,\eta,\Delta,\varepsilon,S)$ is an involutory Hopf algebra in a symmetric monoidal category ${\mathcal{S}}$ with symmetry $\tau$. We derive a topological invariant of closed oriented 3-manifolds by evaluating the morphisms using so-called good pairs of $A$ that we first define.

Good pairs {#sect-def-good-pair}
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A *good pair* for $A$ is a pair $(\phi{\colon}A \to I,\, \Omega{\colon}I \to A)$ of morphisms in ${\mathcal{S}}$, where 