"I am cutting tomatoes with a metal knife."
Translation:Kroję pomidory metalowym nożem.
Why do we not use "z" for "with" in this case? Is it somehow implied in the verb for to cut?
If you use z, it will mean more or less that you are cutting tomatoes while having a knife in your hand. Here you want to say that you are actually using the knife to cut the tomatoes, and for that you simply use the instrumental case without any preposition (that's where the name "instrumental" comes from).
No, it's included in the instrumental case. The basic function of the instrumental case is to tell what was used to perform an action (the instrument). No need to have a preposition if you have that case.
I'm polish guy and in our language we can say "nożem metalowym" and "metalowym nożem".