positive integer $p$ and define the following circles on ${\mathbb{T}}$: $$u={\mathbb{R}}(1,1/p) + {\mathbb{Z}}^2 {\quad \text{and} \quad}l={\mathbb{R}}(0,1) + {\mathbb{Z}}^2.$$ Then $D_p=({\mathbb{T}},\{u\},\{l\})$ is a Heegaard diagram which represents the lens space $L(p,1)$. Clearly, it is ordered (since the sets of upper/lower circles are singletons). We provide the circles $u$ an $l$ with the orientation induced by the standard orientation of ${\mathbb{R}}$. We endow $u$ with the base point $(1/2,1/(2p))$ and $l$ with the base point $(0,1/2p)$. Then $D_p$ is an ordered oriented based Heegaard diagram with $p$ intersection points which are all positive The permutation associated to $D_p$ is the identity and so we have $P_{(1 2 \cdots p)}={\mathrm{id}}_{A^{\otimes p}}$. Then $$K_A(D)=\mu_p \,\Delta_p =\;{\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{KAD-Ln1.eps}}}} \;.$$

3\) Let us consider the following oriented based Heegaard diagram $D=(\Sigma,{\mathcal{U}},{\mathcal{L}})$ of genus 2 used by Poincaré to define his celebrated homology 3-sphere (see [@Po]): $$\psfrag{1}[Bc][Bc]{\scalebox{.9}{$1$}}
\psfrag{2}[Bc][Bc]{\scalebox{.9}{$2$}}
\psfrag{3}[Bc][Bc]{\scalebox{.9}{$3$}}
\psfrag{4}[Bc][Bc]{\scalebox{.9}{$4$}}
\psfrag{5}[Bc][Bc]{\scalebox{.9}{$5$}}
\psfrag{6}[Bc][Bc]{\scalebox{.9}{$6$}}
\psfrag{7}[Bc][Bc]{\scalebox{.9}{$7$}}
\psfrag{a}[Bc][Bc]{\scalebox{.9}{$a$}}
\psfrag{b}[Bc][Bc]{\scalebox{.9}{$b$}}
\psfrag{c}[Bc][Bc]{\scalebox{.9}{$c$}}
\psfrag{d}[Bc][Bc]{\scalebox{.9}{$d$}}
\psfrag{e}[Bc][Bc]{\scalebox{.9}{$e$}}
\psfrag{l}[Bc][Bc]{\scalebox{.9}{$l_1$}}
\psfrag{k}[Bc][Bc]{\scalebox{.9}{$l_1$}}
\psfrag{x}[Bc][Bc]{\scalebox{.9}{$l_2$}}
\psfrag{y}[Bc][Bc]{\scalebox{.9}{$l_2$}}
\psfrag{u}[Bc][Bc]{\scalebox{.9}{$u_1$}}
\psfrag{v}[Bc][Bc]{\scalebox{.9}{$u_2$}}
\psfrag{S}[Bc][Bc]{\scalebox{.9}{$\Sigma$}}
D= \;{\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{HD-Poincare.eps}}}} \;.$$ This pictures represents the surface $\Sigma$ cut along the lower circles $l_1$ and $l_2$. The arrows and the dots represent the orientations and base points of the circles. We order the sets ${\mathcal{U}}=\{u_1,u_2\}$ and ${\mathcal{L}}=\{l_1,l_2\}$ by setting $u_1<u_2$ and $l_1<l_2$. 