More intrinsically knotted graphs

Foisy, Joel

doi:10.1002/jgt.20200

Cited by 6 publications

(6 citation statements)

References 8 publications

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“…Their work attracted attention from scholars in low‐dimensional topology, because a column supporting their proof stands on some firm results from low‐dimensional topology [7,17,18]. Currently the characterization of knotless graphs seems to be far from completion, although numerous minimal forbidden graphs have been discovered until recent years [2–6,9,11,12,14]. In this paper we study another intrinsic property which is derived from embeddings of simple graphs.…”

Section: Introductionmentioning

confidence: 90%

“…, and → 6 7, and some isotopic moves in 3 , we obtain an embedded bouquet graph Θ as illustrated in Figure 18B.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Let G be a planar graph. An embedding → f G : 3 or the embedded graph f G ( ) is said to be trivial, if there exists a topological 2-sphere which contains f G ( ) in 3 .…”

Section: Acknowledgmentsmentioning

confidence: 99%

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Linearly free graphs

Huh

Lee

2021

Journal of Graph Theory

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In this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3‐space R 3. An embedding of a graph into R 3 is said to be linear, if it sends every edge to be a line segment. And we say that an embedding f of a graph G into R 3 is free, if π 1 ( R 3 − f ( G ) ) is a free group. Lastly a simple connected graph is said to be linearly free if every its linear embedding is free. In the 1980s it was proved that every complete graph is linearly free, by Nicholson. In this paper, we develop Nicholson's arguments into a general notion, and establish a sufficient condition for a linear embedding to be free. As an application of the condition we give a partial answer for a question: How much can the complete graph K n be enlarged so that the linear‐freeness is preserved and the clique number does not increase? And an example supporting our answer is provided. As the second application it is shown that a simple connected graph of minimal valency at least 3 is linearly free, if it has less than eight vertices. The conditional inequality is strict, because we found a graph with eight vertices which are not linearly free. It is also proved that for n , m ≤ 6 the complete bipartite graph K n , m is linearly free.

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Section: Introductionmentioning

confidence: 90%

“…, and → 6 7, and some isotopic moves in 3 , we obtain an embedded bouquet graph Θ as illustrated in Figure 18B.…”

Section: Acknowledgmentsmentioning

confidence: 99%

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Linearly free graphs

Huh

Lee

2021

Journal of Graph Theory

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“…Foisy [15] proved that K 3,3,1,1 indeed is IK. His technique, partially outlined below, led to finding many more MMIK graphs later on [16,17,19].…”

Section: Ik Graphsmentioning

confidence: 99%

On intrinsically knotted and linked graphs

Naimi

2020

Preprint

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“…Robertson, Seymour and Thomas [20] gave a Kuratowski-type classification of intrinsically linked graphs, showing that every such graph contains one of the graphs in the Petersen family as a minor (see Figure 1). There is, as yet, no such classification for intrinsically knotted graphs; and since there are dozens of known minor-minimal intrinsically knotted graphs (see [11,12,17]), any such classification will be far more complex.…”

Section: Introductionmentioning

confidence: 99%

Counting Links and Knots in Complete Graphs

Abrams¹,

Mellor²,

Trott³

2013

Tokyo J. Math.

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We investigate the minimal number of links and knots in embeddings of complete partite graphs in S 3 . We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of K 4,4,1 is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.

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More intrinsically knotted graphs

Abstract: Abstract:We demonstrate four intrinsically knotted graphs that do not contain each other, nor any previously known intrinsically knotted graph, as a minor.

Cited by 6 publications

References 8 publications

Linearly free graphs

Linearly free graphs

On intrinsically knotted and linked graphs

Counting Links and Knots in Complete Graphs

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