"I do not know their names."
Translation:Neznám jejich jména.
Czech has two verbs for "know": vědět (vím, víš, ví, víme, víte, vědí) and znát (znám, znáš, zná, známe, znáte, znají). The difference is in usage:
vědět takes a subordinate clause, while znát takes a direct object. So for example:
Znám jeho jméno. (I know his name.) // Vím, jak se jmenuje. (I know what his name is.)
Znám ji. (I know her.) // Vím, kdo to je. (I know who he/she/that is.)
Známe tě dobře. (We know you well.) // Víme, jaký jsi. (We know what you're like.)
A few exceptions: vědět can take "to" (it/that) as an object. There's a subtle difference between "Vím to." (I know that fact or piece of information -as in I know about the thing that was just mentioned) and "Znám to." (I know that object, OR I know something from my personal experience.) Vědět can also take "málo" (little), "hodně" (much), "nic" (nothing) or "všechno" (everything) as an object, especially when combined with "o +locative" (about something).
You only have two "units" of speech here: "neznám" and "jejich jména", which only gives you two options, both valid: "Neznám jejich jména" and "Jejich jména neznám".
The possible inclusion of the stressed pronoun "já" gives you three more options - it would likely be at the beginning: "Já neznám jejich jména", "Já jejich jména neznám", or - and this is already as quirky as we can get without sounding too ridiculous: "Jejich jména já neznám" (something akin to: As for their names, I surely don't know them.).
In standard langauge, we cannot separate "jejich jména", it's a fixed unit. We can only do that in poetry – then you get more combinations, all sounding highly unnatural in normal speech, such as: "Jejich neznám jména" or "Neznám jmen jejich" (I changed the object from the accusative into the genitive to give it an extra poetic sound here.) But please don't expect Duolingo to accept such poetic constructs ;)