Is it travelling if a player does not have his dribble and touches the floor with a hand or knee, but actually keeps pivot foot solidly on the floor?

Hand? No.

Knee? Yes.

4-42-5 (the reference may be different in this year's book, but the rule's the same): A player holding the ball may not touch the floor with a knee or any other part of te body other than hand or foot.

4.43.5 . . . A player holding the ball:
a. May not touch the floor with a knee or any other part of the body other than hand or foot.
b. After gaining possession while on the floor and touching with other than hand or foot, may not attempt to get up or stand.