Can’t give you sources but what i was raised with was-the original challah on the Shulchan was braided, that is why we do it. The reason it was braided was to show unity – all the different strands are braided into ONE unit. For Rosh Hashona we don’t braid. (although it seems like there are many people who do – i was raised that it shouldn’t have braids) The reason was that we should have a Smooth year with no knots/lumps. May it be so for all of klal yisroel
