Maximal knotless graphs

Eakins, Lindsay; Fleming, Thomas; Mattman, Thomas

doi:10.2140/agt.2023.23.1831

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Maximal knotless graphs

Lindsay Eakins, Thomas Fleming and Thomas Mattman

Algebraic & Geometric Topology 23 (2023) 1831–1848

DOI: 10.2140/agt.2023.23.1831

Abstract

A graph is maximal knotless if it is edge maximal for the property of knotless embedding in $ℝ^{3}$ . We show that such a graph has at least $\frac{7}{4} | V |$ edges, and construct an infinite family of maximal knotless graphs with $| E | < \frac{5}{2} | V |$ . With the exception of $| E | = 22$ , we show that for any $| E | \geq 20$ 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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Volume 23, issue 4 (2023)