Slice the end off the cucumber then cut into quarters lengthways and remove seeds. Thinly slice each quarter lengthways to make spears.
Combine the mayonnaise and chopped crabsticks.
Lay a sheet of nori on a bamboo mat. With wet hands, pat a thin, even layer of rice over the nori, leaving a 1cm edge at the bottom. Sprinkle rice with sesame seeds. Carefully flip over.
About 1/3 of the way from the top of the nori sheet, add 1 tablespoon of the crab mixture in a line across the sheet. Layer with 1 to 2 cucumber spears and 1 to 3 slices of avocado.
Using the bamboo mat, roll the sheet tightly into a cylinder, starting at the top edge. Once at the bottom, moisten the uncovered edge with water and finish rolling. Repeat with remaining three sheets of nori.
Using a wet knife, slice each roll into 2cm pieces. Garnish each piece with 1/8 teaspoon of tobiko.