Heegaard diagrams of lens spaces

Abstract:

Let $(M,F;\upsilon ,w)$ be a Heegaard diagram of $M$. The complexity of this diagram is the number of points in $\upsilon \cap m$. This is also the length of the relators in a group presentation naturally corresponding to this diagram. We give an example to show that a Heegaard diagram of minimal complexity need not have a cancelling pair of meridian disks. In terms of the presentation, this says that a minimal length presentation need not have a defining relator for one of the generators. This provides a counterexample to a conjecture of Waldhausen. Our example depends on the rather trivial observation that the shortest possible $2$-generator presentation of the cyclic group of order $173$ is $\left \langle {a,b|{a^{13}}{b^2},{a^{ - 2}}{b^{13}}} \right \rangle$.