The tuber will eventually rot and make a stinky mess so you can really detatch the entire vine with roots intact and replant it in a pot with potting soil if you like.

Thank you so much!

The green stems are slips and can be transplanted once they have at leaf one, but preferably 2 leaf nodes. You might need to keep the new soil a little damp for the first day and check it the first week as it gets established, but it will produce many leaves and vines.