Slice the end off of the cucumber, then cut into quarters lengthways. 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.
Slice each roll into 2cm pieces. Garnish each piece with 1/8 teaspoon or tobiko.