Pairs of knot invariants

Let α be a map from the set of all knot types K to a set X. Let β be a map from K to a set Y. We define the relation between α and β to be the image of a map (α,β) from K to X× Y sending an element K of K to (α(K),β(K)). We determine the relations between α and β for certain α and β such as crossing number, unknotting number, bridge number, braid index, genus and canonical genus.