In a medium bowl, mix together the crabmeat, mayonnaise, lemon juice, and sugar.
Place a piece of plastic wrap on the sushi mat. Using wet fingers, spread 1 1/4 cups of the rice evenly over the plastic wrap to form a rectangle slightly smaller than the sushi mat. Lay a sheet of nori over the rice. Arrange 1/4 of the crab mixture along the center of the nori leaving a 2-inch border at the top. Top the crab mixture with 1/4 of the avocado slices, 1/4 of the mango slices, and 1/4 of the red bell pepper slices. Using the sushi mat as a guide, fold the rice over the filling and roll into a cylinder. Remove the plastic wrap. Repeat with the remaining ingredients. Sprinkle each roll with the black sesame seeds.
To serve, using a sharp, damp knife, slice each cylinder into 6 pieces (re-wet the knife after each slice). Arrange the crab rolls on a serving platter and serve with wasabi, sweet pickled ginger and soy sauce.
*Can be found at specialty Asian markets
Place the rice and water in a medium saucepan over high heat. Bring the mixture to a boil. Cover the pan and reduce the heat to low. Simmer for 25 minutes. Sprinkle the vinegar over the rice and mix with a fork. Transfer the cooked rice to a parchment paper-lined baking sheet. Allow the rice to cool completely before using.
For spicier rolls, substitute the red bell pepper with 2 seeded and thinly sliced jalapenos.