Pairs of knot invariants

Let Formula: see text be a map from the set of all knot types Formula: see text to a set Formula: see text. Let Formula: see text be a map from Formula: see text to a set Formula: see text. We define the relation between Formula: see text and Formula: see text to be the image of a map Formula: see text from Formula: see text to Formula: see text sending an element Formula: see text of Formula: see text to Formula: see text. We determine the relations between Formula: see text and Formula: see text for certain Formula: see text and Formula: see text such as crossing number, unknotting number, bridge number, braid index, genus and canonical genus. This is a study of geography problem in knot theory.