on the set ${\mathcal{I}}$ inherited from ${\mathcal{L}}$.

Tensors associated to Heegaard diagrams using Hopf algebras {#sect-asso-tensor-HD}
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Let $A$ be a Hopf algebra in a symmetric monoidal category ${\mathcal{S}}$. Let $D=(\Sigma,{\mathcal{U}},{\mathcal{L}})$ be an ordered oriented based Heegaard diagram. Following Kuperberg [@Ku1], we associate to $D$ a morphism in ${\mathcal{S}}$ $$K_A(D) {\colon}A^{\otimes g} \to A^{\otimes g},$$ where $g$ is the genus of $\Sigma$, as follows. Denote by ${\mathcal{I}}$ the set of intersection points of $D$. By Section \[sect-opoHD\], the set ${\mathcal{I}}$ inherits two total orders from ${\mathcal{U}}$ and ${\mathcal{L}}$. Denote by ${\mathcal{I}}_{\mathcal{U}}$ (respectively ${\mathcal{I}}_{\mathcal{L}}$) the set ${\mathcal{I}}$ endowed with the order induced by ${\mathcal{U}}$ (respectively ${\mathcal{L}}$). Let $N$ be the cardinality of ${\mathcal{I}}$. Consider the permutation $\sigma \in \mathfrak{S}_N$ such that if an intersection point is the $i$-th element of ${\mathcal{I}}_{\mathcal{L}}$, then it is the $\sigma(i)$-th element of ${\mathcal{I}}_{\mathcal{U}}$. By Section \[sect-repres-Sn\], we derive from $\sigma$ and the symmetry of ${\mathcal{S}}$ an automorphism $$P_\sigma {\colon}A^{\otimes N} \to A^{\otimes N}.$$ Set $$S_{\mathcal{L}}=\bigotimes_{c \in {\mathcal{I}}_{\mathcal{L}}} S^{\kappa_c}  {\colon}A^{\otimes N} \to A^{\otimes N}  {\quad \text{and} \quad}S_{\mathcal{U}}=\bigotimes_{c \in {\mathcal{I}}_{\mathcal{U}}} S^{\kappa_c}  {\colon}A^{\otimes N} \to A^{\otimes N},$$ where $S$ is the antipode of $A$ and $$\kappa_c=
\begin{cases}
0 & \text{if $c$ is positive,} \\ 1 & \text{if $c$ 