I'm trying to understand the chemical formula of soap using potash ($\ce{K2CO3}$) as they did in antiquity instead of using sodium hydroxide ($\ce{NaOH}$) or potassium hydroxide ($\ce{KOH}$). I've found the formula for making soap with sodium hydroxide on the Khan Academy, but I didn't make much progress on my own.
Can somebody help me understand how the soap and the glycerin molecule would look like?
Or is the saponification not possible and ancient soap makers first needed to create potassium hydroxide from potassium carbonate?