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

Abstract

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.