NP–hard problems naturally arising in knot theory

We prove that certain problems naturally arising in knot theory are NP–hard or NP–complete. These are the problems of obtaining one diagram from another one of a link in a bounded number of Reidemeister moves, determining whether a link has an unlinking or splitting number k k , finding a k k -component unlink as a sublink, and finding a k k -component alternating sublink.