When I was a kid, the response to why are YOU cutting cheese used to be 'he who smelteth, dealteth' (he who smelled it, did it). So now I'm imagining a surreal conversation in my kitchen, in which I'm asked why I am cutting real cheese and accuse the other person of cutting cheese because he's the one who smelled it.
Lost, so lost, in translation...
By the way, I once asked a student what his surname - Bąk - meant. There was an embarrassed silence among my advanced Polish learners of English. Finally someone said, 'Something like the buzzing of a bee,' and there were suppressed giggles and snorts. Someone explained to me later that it's a child's word for... cutting the cheese (cue sound effect).
Polish surnames can be an source of endless speculation and mirth. I mean, I can understand someone getting a surname like 'Carpenter' (someone was Jim, the carpenter) or 'North' (Jim who lives at the north end of the village) or even 'Lamb' (Jim who helps at lambing time, perhaps). But how do you end up with surnames like Cuckoo and RUN! (in the imperative)? I once knew a hopelessly lazy, slow and inefficient receptionist whose surname was... Czekaj. It's like living in bad fiction.
Well, I explained the first two above, I believe: "kroić" is what you do with a knife: cutting/slicing bread, meat, vegetables, "ciąć" is what you do with scissors, so mostly cutting paper.
Also "ciąć" has an irregular declension: ja tnę, ty tniesz, on/ona/ono tnie, my tniemy, wy tniecie, oni/one tną.
"pokroić" is perfective. So "pokroiłem mięso" is "I cut the meat" (I finished, it is now as I wanted it to be). "kroiłem mięso" is like "I was cutting the meat", it just talks about the process.
Wow, I never paid attention to the comments here and I haven't been aware that it's something idiomatic in English until now... English never ceases to surprise me :D This sentence can only be interpreted literally in Polish.
OK, it's not a vulgar word so I guess there's no harm in answering your question... generally, "to fart" = "pierdnąć" (perfective) and "pierdzieć" (imperfective). You'd find it in a dictionary anyway. As for something idiomatic, I think the most common is "puścić bąka" (lit. to let go a bumblebee).
Thank you Alik, I but I beg to differ, in English cheese is countable. Last month in Parma, I bought two cheeses. https://en.oxforddictionaries.com/definition/cheese
In Pl. it is also countable, btw https://wsjp.pl/index.php?id_hasla=23056&id_znaczenia=4580778&l=23&ind=0