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

Abstract : We study the class L of link types that admit a K 4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K 4. 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 K 4-minor-free link diagrams, all minimal connected K 4-minor-free link diagrams, and all K 4-minor-free diagrams of the unknot.
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Contributor : Dimitrios M. Thilikos <>
Submitted on : Monday, October 8, 2018 - 4:33:48 PM
Last modification on : Thursday, November 22, 2018 - 12:13:15 PM
Long-term archiving on : Wednesday, January 9, 2019 - 4:36:54 PM

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Juanjo Rué, Dimitrios M. Thilikos, Vasiliki Velona. Structure and Enumeration of $K4$-minor-free links and link diagrams. Electronic Notes in Discrete Mathematics, Elsevier, 2018, 68, pp.119-124. ⟨10.1016/j.endm.2018.06.021⟩. ⟨lirmm-01890505⟩

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