Prospects are not automatically added to your Link Partners and Links list. Prospects are only converted to Link Partners and Links when you explicitly approve them. Also, once a prospect is converted to Link Partner and Link it will no longer show up in the Prospects list the next time you view that specific Prospecting Profile.
All Prospect search results are isolated in the Prospecting module within the project you created the Prospecting Profile in. When you select a Prospecting search you will see the current filter set to "Opportunities for" with the Prospecting search name following in "quotes".