Player A1 jumps up for a shot under the basket. Player B1 blocks the ball cleanly so that A1 returns to the floor with the ball, the ball never leaving his hands. Jump ball or travel?

Does it matter if on the same play if the block causes the ball to leave A1's hands before he catches it again and returns to the floor?

Did both the shooter and jumper have there hands on the ball at the same time, if so, I would call a jump ball. If they both came down with the ball, its a jump. But if the defensive play blocks it and the shooter returns to the floor with the ball, nothing.

If the "block" prevents the release, it's a held ball.

If the "block" knocks the ball loose, play on.

If A1 "chooses" not to release the ball, or "chooses" to release it and then gathers it again, it's a violation.

99% of the time, it's either of the first two options.

I agree with bob, most of the time its the first two.