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 $L\in L$. 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.