Amusingly, the word anatomy (“dissection”) is from Greek via Latin, from the very same root that we’re discussing here.
Other English terms from the Greek root for “cut”: tomography (imaging through a lot of cross-sections); entomology (study of in-sects, critters with sect-ions in their bodies); dichotomy (division into two possibilities); atom (that which cannot be divided).
“Up” or “thoroughly”, apparently? Same prefix as in analysis, anaphora, anamorphism. Ancient Greek had a sprawling system of prefixes that one can’t really pick up by osmosis, it seems.
(Complaints about noun morphology sound a bit hypocritical from a native speaker of Russian, I know, but it is what it is.)