Structure and Enumeration of K 4 -minor-free links and link diagrams

Abstract

We study the class L of link-types that admit a K 4 -minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K 4 . We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L (and subclasses of it), with respect to the minimum number of crossings or edges in a projection of L ∈ L. Further, we obtain counting formulas and asymptotic estimates for the connected K 4 -minor-free link-diagrams, minimal K 4 -minor-free link-diagrams, and K 4 -minor-free diagrams of the unknot.