Pairs of knot invariants

Abstract

Let $\alpha$ be a map from the set of all knot types ${\mathcal K}$ to a set $X$. Let $\beta$ be a map from ${\mathcal K}$ to a set $Y$. We define the relation between $\alpha$ and $\beta$ to be the image of a map $(\alpha,\beta)$ from ${\mathcal K}$ to $X\times Y$ sending an element $K$ of ${\mathcal K}$ to $(\alpha(K),\beta(K))$. We determine the relations between $\alpha$ and $\beta$ for certain $\alpha$ and $\beta$ such as crossing number, unknotting number, bridge number, braid index, genus and canonical genus. This is a study of geography problem in knot theory.