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Maximal knotless graphs

Lindsay Eakins, Thomas Fleming and Thomas Mattman

Algebraic & Geometric Topology 23 (2023) 1831–1848
Abstract

A graph is maximal knotless if it is edge maximal for the property of knotless embedding in 3. We show that such a graph has at least 7 4|V | edges, and construct an infinite family of maximal knotless graphs with |E| < 5 2|V |. With the exception of |E| = 22, we show that for any |E| 20 there exists a maximal knotless graph of size |E|. We classify the maximal knotless graphs through nine vertices and 20 edges. We determine which of these maxnik graphs are the clique sum of smaller graphs and construct an infinite family of maxnik graphs that are not clique sums.

Keywords
maximal knotless graph, spatial graphs, intrinsic knotting, knotless embedding, maximal planar
Mathematical Subject Classification
Primary: 05C10
Secondary: 57K10, 57M15
References
Publication
Received: 12 January 2021
Revised: 26 August 2021
Accepted: 30 November 2021
Published: 14 June 2023
Authors
Lindsay Eakins
Department of Mathematics and Statistics
California State University, Chico
Chico, CA
United States
Thomas Fleming
New York, NY
United States
Thomas Mattman
Department of Mathematics and Statistics
California State University at Chico
Chico, CA
United States

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