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.

32 pages, 23 figures, things related to geography and botany problems in mathematics added