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### More intrinsically knotted graphs

Joel Foisy
2006 Journal of Graph Theory
We demonstrate four intrinsically knotted graphs that do not contain each other, nor any previously known intrinsically knotted graph, as a minor.  ...  They thus lead, by ∆ − Y exchanges, to even more intrinsically knotted graphs  . We will not further discuss these other graphs in this paper.  ...  With a little more work, we can establish the following: Theorem 3.3. The graph J 14 is intrinsically knotted. Proof. Embed J 14 .  ...

### Many, many more intrinsically knotted graphs

Noam Goldberg, Thomas W Mattman, Ramin Naimi
2014 Algebraic and Geometric Topology
We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.  ...  A graph with this property is said to be intrinsically knotted (IK). The question "Which graphs are IK?" has remained open for the past 30 years.  ...  It is easy to show that a graph is IK if and only if it contains a minor that is minor minimal intrinsically knotted (MMIK).  ...

### More intrinsically knotted graphs with 22 edges and the restoring method [article]

Hyoungjun Kim, Thomas Mattman, Seungsang Oh
2017 arXiv   pre-print
A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle.  ...  Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges.  ...  The ∇Y move preserves intrinsic knottedness  . Sachs  observed this for intrinsic linking and the proof for intrinsic knotting is similar.  ...

### Intrinsic knotting and linking of almost complete partite graphs [article]

Thomas W. Mattman, Ryan Ottman, Matt Rodrigues
2004 arXiv   pre-print
In addition, we classify intrinsic knotting of graphs on 8 vertices.  ...  For graphs in these families, we verify a conjecture presented in Adams' "The Knot Book": If a vertex is removed from an intrinsically knotted graph, one obtains an intrinsically linked graph.  ...  is not intrinsically knotted and has more than 5n − 15 edges?  ...

### An Introduction to Virtual Spatial Graph Theory [article]

Thomas Fleming, Blake Mellor
2006 arXiv   pre-print
Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs.  ...  Acknowledgement: The authors would like to thank Akio Kawauchi and the organizers of the conference Knot Theory for Scientific Objects for giving them the opportunity to introduce this work.  ...  This produces a filtration of the set of intrinsically linked graphs, and the deeper in the filtration a graph persists, the more "linked" the graph is.  ...

### A sufficient condition for intrinsic knotting of bipartite graphs [article]

Sophy Huck, Alexandra Appel, Miguel-Angel Manrique, Thomas W Mattman
2008 arXiv   pre-print
Finally, we classify bipartite graphs with ten or fewer vertices with respect to intrinsic knotting.  ...  We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted.  ...  Indeed, by the inductive hypothesis, all K n−1,an \ (n − 1 − 4)a n − a n−1 graphs are IK. Then, applying Lemma 6, it follows that all K n,an \ (n − 5)a n graphs are IK, as required.  ...

### Knotting and linking in the Petersen family [article]

Danielle O'Donnol
2010 arXiv   pre-print
This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs.  ...  We also present a more explicit relationship for the graph K_3,3,1 between knotting and linking, which relates the sum of the squares of linking numbers of links in the embedding and the second coefficient  ...  The set of intrinsically knotted graphs has not been fully characterized, however it is known that every intrinsically knotted graph is intrinsically linked.  ...

### Intrinsic knotting and linking of almost complete graphs [article]

J. Campbell, T.W. Mattman, R. Ottman, J. Pyzer, M. Rodrigues, S. Williams
2007 arXiv   pre-print
In addition, we classify intrinsic knotting of graphs on 8 vertices.  ...  A graph on n vertices with at least 5n-14 edges is intrinsically knotted.  ...  Thus any subgraph of K 8 obtained by removing 8 or more edges is not intrinsically knotted. In total then, there are twenty intrinsically knotted graphs on 8 vertices.  ...

### Bipartite intrinsically knotted graphs with 22 edges [article]

Hyoungjun Kim, Thomas Mattman, Seungsang Oh
2014 arXiv   pre-print
A graph is intrinsically knotted if every embedding contains a knotted cycle.  ...  It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, K_7 and the 13 graphs obtained from K_7 by ∇ Y moves, are the only minor minimal intrinsically knotted graphs  ...  Suppose that G is an intrinsically knotted graph with 22 edges that has a vertex a in A of degree 6 or more.  ...

### Intrinsically knotted graphs

Joel Foisy
2002 Journal of Graph Theory
We demonstrate four intrinsically knotted graphs that do not contain each other, nor any previously known intrinsically knotted graph, as a minor.  ...  They thus lead, by ∆ − Y exchanges, to even more intrinsically knotted graphs  . We will not further discuss these other graphs in this paper.  ...  With a little more work, we can establish the following: Theorem 3.3. The graph J 14 is intrinsically knotted. Proof. Embed J 14 .  ...

### Recent Developments in Spatial Graph Theory [article]

Erica Flapan, Thomas Mattman, Blake Mellor, Ramin Naimi, Ryo Nikkuni
2016 arXiv   pre-print
In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs.  ...  This article presents a survey of some recent results in the theory of spatial graphs.  ...  We have the following more general results about the negative relationship between n-apex and intrinsic linking and knotting. ,  No intrinsically knotted graph is 2-apex.  ...

### Intrinsic linking and knotting in virtual spatial graphs

Thomas Fleming, Blake Mellor
2007 Algebraic and Geometric Topology
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs.  ...  We also provide several examples of intrinsically virtually linked and knotted graphs.  ...  V K : V K . p 1/ D . 1/ Arf.K / [ The same argument shows that K 3;3;1;1 and Foisy's graph H are also intrinsically virtually knotted of degree 1, since the intrinsic knottedness of both graphs was proved  ...

### Most Graphs are Knotted [article]

Kazuhiro Ichihara, Thomas W. Mattman
2018 arXiv   pre-print
We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases.  ...  We also argue that, for k ≥ 18, most graphs of order k are intrinsically knotted and, for k ≥ 2n+9, most of order k are not n-apex.  ...  Thus, a graph of order seven or more with at least 5|G| − 14 edges is IK.  ...

### Page 4270 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews
More precisely, they prove that Ko is intrinsically triple linked, but Ko is not intrinsically triple linked.  ...  For example, Kg (the complete graph on six vertices) is intrinsically linked and K7 is MANIFOLDS AND CELL COMPLEXES 4270 intrinsically knotted.  ...

### Intrinsic linking and knotting in tournaments [article]

Thomas Fleming, Joel Foisy
2019 arXiv   pre-print
We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties.  ...  We produce the following bounds: intrinsically linked (n=8), intrinsically knotted (9 ≤ n ≤ 12), intrinsically 3-linked (10 ≤ n ≤ 23), intrinsically 4-linked (12 ≤ n ≤ 66), intrinsically 5-linked (15 ≤  ...  T not intrinsically knotted as a directed graph.  ...
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