  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' in 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.
