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

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...

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Bibliographic Details
Published in:European journal of combinatorics 2020-10, Vol.89, Article 103147
Main Authors: Rué, Juanjo, Thilikos, Dimitrios M., Velona, Vasiliki
Format: Article
Language:English
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Summary: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∈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.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2020.103147