Rinse the sushi rice under cold water until the water runs clear.
Combine the rice and water in a rice cooker and cook according to the manufacturer's instructions.
In a small saucepan, heat rice vinegar, sugar, and salt until dissolved.
Once the rice is cooked, transfer it to a large bowl and gently fold in the vinegar mixture.
Allow the rice to cool to room temperature.
Place a sheet of nori on a bamboo sushi mat, shiny side down.
Wet your hands and spread a thin layer of rice over the nori, leaving a 1-inch border at the top.
Arrange avocado, cucumber, and crab in a line across the center of the rice.
Using the mat, roll the sushi tightly away from you, pressing gently to secure.
Slice the roll into bite-sized pieces and sprinkle with sesame seeds.