As you can see from the picture, in the top of the fingers, there are various tendons that control finger movement. If your cut is deep, it may cut the tendons, making your unable to move your index finger. See a hand surgeon to fix it.
Do you mean you cannot move your finger or the distal joint? If yes, you may have damaged your finger tendon or nerve. You are suggested to go to see an orthopedist for treatment.