Structure and enumeration of K₄-minor-free links and link-diagrams

We study the class L of link-types that admit a K₄-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K₄. 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 K₄-minor-free link-diagrams, minimal K₄-minor-free link-diagrams, and K₄-minor-free diagrams of the unknot.