Let $S$ be a torus with one boundary component. The group $H_1(S)$ equals $\mathbb{Z}^2$. Fixing some such identification, $(a,b) \in \mathbb{Z}^2 = H_1(S)$ can be represented by a simple closed curve iff and only if $a$ and $b$ are relatively prime.

Let $v_0$ be a point on $\partial S$, and let $\alpha, \beta \in \pi_1(S,v_0)$ be elements with the following properties.

- $\{\alpha,\beta\}$ is a free basis for $\pi_1(S,v_0)$.
- $[\alpha] = (1,0) \in H_1(S)$ and $[\beta] = (0,1) \in H_1(S)$.
- $\alpha$ and $\beta$ can be represented by based simple closed curves that only intersect at the basepoint $v_0$.

If you draw the picture, this is just the standard way of drawing a basis for $\pi_1(S,v_0)$.

This brings me to my question. Fix $(a,b) \in H_1(S)$ such that $a$ and $b$ are relatively prime. What is the shortest word $w$ in $\{\alpha,\beta\}$ such that $[w] = (a,b)$ and $w$ can be represented by a based simple closed curve?