= Solution
Use the <meridian of a knot> $\mu$ and <Seifert longitude> $\lambda$ of the <unknot> on its boundary <torus>. The <unknot>'s <knot exterior> is a <solid torus> whose disk-bounding curve is $\lambda$. A genus-one <Heegaard diagram> is therefore
$$
\boxed{\Sigma=\mathbb R^2/\mathbb Z^2,\qquad \alpha=\lambda=(0,1),\qquad \beta=5\mu+3\lambda=(5,3).}
$$
Here the horizontal coordinate is $\mu$ and the vertical coordinate is $\lambda$. The drawing uses a small translate of $\alpha$ to avoid intersections on the edge of the square; opposite edges are identified. The <algebraic intersection number of curves on an oriented surface> has absolute value five.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-141-lens.png]
{title=A Heegaard torus for a <lens space>}
Starting with $\Sigma\times[0,1]$, attach a three-dimensional <two-handle> along $\alpha\times\{0\}$, using its <surface framing>, and cap the resulting <sphere> with a <three-handle>. This gives the <solid torus> on that side. Attach a <two-handle> along $\beta\times\{1\}$, again using the <surface framing>, and cap its <sphere> with a <three-handle>. The second <solid torus> has its disk-bounding curve identified with $5\mu+3\lambda$, so the resulting oriented <lens space> is precisely the specified $5/3$ <Dehn filling>.
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