definition 0.0.1 Let (Σg,{αi}i=1g,{βi}i=1g) be a Heegaard diagram. We say that another diagram (Σg,{αi′}i=1g,{βi′}i=1g) is related to (Σg,{αi}i=1g,{βi}i=1g) by an isotopy if the sets {αi}i=1g,{αi′}g are isotopic in Σg, or {βi}i=1g,{βi′}g are isotopic in Σg. handle slide if αi=αi′ for i≠g, and the curves αg−1,αg,αg′ bound a pair of pants in Σg disjoint from {αi}i=1g−2 (or similarly for the βi). See (handleslide)