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.