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 minimal number of crossings or edges in a projection of L∈L. Further, we enumerate (both exactly and asymptotically) all connected K₄-minor-free link diagrams, all minimal connected K₄-minor-free link diagrams, and all K₄-minor-free diagrams of the unknot.