$A^*$ are $$\psfrag{B}[Bl][Bl]{\scalebox{.9}{$A^*$}}
\psfrag{c}[Bc][Bc]{\scalebox{.8}{${\mathrm{coev}}_A$}}
\psfrag{a}[Bc][Bc]{\scalebox{.8}{${\mathrm{ev}}_A$}}
\mu_{A^*}=\, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{mu-Adual.eps}}}}\,, \qquad \Delta_{A^*}= \, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{delta-Adual.eps}}}}\,,$$ and the unit, counit, and antipode of $A^*$ are $$\psfrag{B}[Bl][Bl]{\scalebox{.9}{$A^*$}}
\psfrag{c}[Bc][Bc]{\scalebox{.8}{${\mathrm{coev}}_A$}}
\psfrag{a}[Bc][Bc]{\scalebox{.8}{${\mathrm{ev}}_A$}}
\eta_{A^*}=\, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{unit-Adual.eps}}}}\,, \qquad   \Delta_{A^*}= \, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{counit-Adual.eps}}}}\,,
\qquad  S_{A^*}= \, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{antipode-Adual.eps}}}} \,.$$

Example of a super Hopf algebra {#sect-ex-super-HA1}
-------------------------------

Let ${\Bbbk}$ be a field. Pick a positive integer $n$ not divisible by the characteristic of ${\Bbbk}$ and an element ${\omega}\in{\Bbbk}$. Consider the ${\Bbbk}$-algebra ${H_{\omega}^n}$ generated by $t$ and ${\theta}$ submitted to the relations $$t^n=1, \quad {\theta}t = t {\theta}, \quad {\theta}^2={\omega}(t^2-1).$$ Note that ${H_{\omega}^n}$ is finite-dimensional (over ${\Bbbk}$) with basis $\{t^k,\, {\theta}t^k\}_{0 \leq k \leq n-1}$ and $t$ is invertible with inverse $t^{-1}=t^{n-1}$. By assigning the degrees $$|t|=0 {\quad \text{and} \quad}|{\theta}|=1,$$ we obtain that ${H_{\omega}^n}$ is an algebra in the symmetric monoidal category $\mathrm{SVect}_{\Bbbk}$ of super vector [$\Bbbk$-]{}spaces (see Section \[sect-ex-super-vect\]). Moreover, it becomes a Hopf algebra in $\mathrm{SVect}_{\Bbbk}$ with coproduct $\Delta$, counit $\varepsilon$, and antipode $S$ given by $$\begin{aligned}
& \Delta (t)=t\otimes t, && \varepsilon(t)=1, && S(t)=t^{-1}, \\
& \Delta({\theta})=t \otimes {\theta}+{\theta}\otimes 1, &&  \varepsilon({\theta})=0, && S({\theta})=-{\theta}t^{-1}.\end{aligned}$$ Since $S^2(t)=t$ and $S^2({\theta})={\theta}$, the Hopf algebra ${H_{\omega}^n}$ in $\mathrm{SVect}_{\Bbbk}$ is involutory.

Heegaard diagrams and their tensors {#sec2}
===================================

Heegaard diagrams {#sect-Heegaard-diag}
-----------------

By a *circle* on a surface 