I have a purple vein growing from where the gum under my bottom teeth connects to my lip. what is this?

Vein?: It sounds like you might be right. It might be a normal vein near the surface of your gums\mucosa. Similar to when people have varicose veins in their legs. See your own dentist to evaluate and confirm this.

Purple vein: The gum is attached to the lip by a fiber called a frenulum. It is not normally purple, so you should have it examined by your dentist.