Simpliest simple closed curve in fund group of one-holed torus representing given homology class

Question

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.

1. $\{\alpha,\beta\}$ is a free basis for $\pi_1(S,v_0)$.
2. $[\alpha] = (1,0) \in H_1(S)$ and $[\beta] = (0,1) \in H_1(S)$.
3. $\alpha$ 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?

Answer 1

Take the straight line from $0$ to $(a, b)$ in the usual integer lattice (think of ruled paper). Every time you intersect a horizontal line, write $\alpha,$ every time you intersect a vertical line, write $\beta$ When you are done, you will have your shortest word -- I might be off by one, so you should look at the paper by Osborne and Zieschang from 1981. For much more on this subject, look at papers by G. McShane and I. Rivin.