Divide the nori sheets in half, along the fold and split. If it doesn't break easily then use scissors. Cover a sushi rolling mat with cling wrap/film (or put inside a freezer bag).
Put the drained crab in a small bowl and mix in the mayonnaise (if you are using crab sticks, they probably won't mix with the mayo, but split the sticks in half lengthwise and spread a little mayo on the cut side, although the mayo is more optional with them).
Cut the cucumber in half lengthwise, then use one of these halves (see notes if you don't have a Persian cucumber). Cut it into four lengths, remove some of the seeds if there are a lot.
Cut long slices from the avocado then drizzle over a little lime juice and rub over the slices carefully to help stop browning.
Put one of the half nori sheets on the covered rolling mat and take roughly half a cup of the rice and press a layer evenly over the pieces of nori, pressing right to the edges. This is typically done (and easiest) by working from one long edge down towards the other, going across the sheet. Sprinkle over some sesame seeds all over then carefully flip it over so the seaweed side faces up.
Lay a piece of cucumber the length of a long edge, near one side, put a line of crab next to it then a row of avocado slices, trying to make each fairly even all along.
Lift up the edge of the rolling mat next to the side where you have the fillings and bring it over the top of the roll so the fillings are rolled inside the seaweed. Press the mat slightly as you do so. Lift up the edge of the mat so you don't wrap it into the roll but continue to roll up the mat with the sushi in it so it rolls on top of itself. Once you have reached the edge of the rice, unwrap the mat and you should have a formed roll.
Transfer the roll to a board and use a sharp knife to cut it in half in the middle. Cut each half into three slices. Repeat with the remaining nori, rice and fillings. Serve as you prefer, but traditionally with some pickled ginger and soy sauce for dipping on the side.