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

TL;DR

This paper characterizes and enumerates link-types that can be represented with diagrams free of K4 minors, revealing their structural composition and providing counting formulas and asymptotic estimates.

Contribution

It establishes that K4-minor-free links are the closure of certain torus links under connected sum and provides enumeration and asymptotic analysis of these links.

Findings

K4-minor-free links are the closure of a subclass of torus links.

Derived counting formulas for K4-minor-free link diagrams.

Provided asymptotic estimates for various subclasses of K4-minor-free links.

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 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.