This question confused me for a long while. It turns out that my initial thought was correct though. Historically "Amn't I" was correct, but it fell out of favour. In English we have very few (I'm struggling to think of an example to be honest) "mn" sounds. We have words spelt with "mn" (for example "column") but we don't pronounce the full "mn" sound. "Amn't" changed into "An't" and then further devolved into "Ain't" and "Aren't". Nowadays "Aren't" is more commonly seen, and "Ain't" is now viewed as more colloquial (so much so that I would recommend that you don't use it, it's only used in some dialects and sounds very out of place in any other form of speech). Interesting to note is that some dialects still use "Amn't", including some places in Ireland, Scotland and the north of England.
The short answer is: "Amn't" was correct hundreds of years ago, but the language has morphed throughout the years and now "Aren't" is the correct and most commonly seen form.
Reference: https://stancarey.wordpress.com/2014/03/04/amnt-i-glad-we-use-amnt-in-ireland/
Please note that this is a personal blog I found whilst researching this question, so this could all be wrong.