No. When you use something as an instrument, you just use bare Instrumental without any preposition. If you add "z", it's as if you animated the instrument, and the knife is there with you, also cutting.
I like the example with eating: "Jem widelcem" (I am eating, using a fork); "Jem z widelcem" (I am eating, and the fork is also eating).
So this joke wouldn't work in Polish:
- Should you eat French fries with your fingers?
- No, you should eat your fingers separately.
I presume that "with your fingers = using your fingers" would be instrumental, and "with your fingers = at the same time as you eat your fingers" would be z + instrumental?