Structure and enumeration of K4-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 minimal number of crossings or edges in a projection of L ∈ L . Further, we enumerate (both exactly and asymptotically) all connected K 4 -minor-free link diagrams, all minimal connected K 4 -minor-free link diagrams, and all K 4 -minor-free diagrams of the unknot.