How is liver connected to lumen of gut?

The liver is not directly connected to the lumen of the gut. There is an indirect connection via the common bile duct.

The common bile duct is a small tube that carries bile from the liver and gallbladder to the duodenum, the first part of the small intestine.

Bile helps to digest fats and certain vitamins.

This bile enters the gut (duodenum) via sphincter of Oddi.