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## Filters

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More intrinsically knotted graphs

2006
*
Journal of Graph Theory
*

We demonstrate four

doi:10.1002/jgt.20200
fatcat:6ls5ht2fz5g6vg6ypmcgvpl7n4
*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*[6] . 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

2014
*
Algebraic and Geometric Topology
*

We list

doi:10.2140/agt.2014.14.1801
fatcat:bn4mgcjfbja4tl4yur3vjxuq5u
*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]

2017
*
arXiv
*
pre-print

A

arXiv:1708.03925v1
fatcat:zgoang67uzcd3o2mhsfj65yypi
*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*[13] . Sachs [16] observed this for*intrinsic*linking and the proof for*intrinsic**knotting*is similar. ...##
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Intrinsic knotting and linking of almost complete partite graphs
[article]

2004
*
arXiv
*
pre-print

In addition, we classify

arXiv:math/0312176v2
fatcat:exikikv7tvfivfasvf5wmrvdyi
*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]

2006
*
arXiv
*
pre-print

Two natural generalizations of

arXiv:math/0608739v1
fatcat:zkaxiasppfc57pinnuxu2nafve
*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]

2008
*
arXiv
*
pre-print

Finally, we classify bipartite

arXiv:0811.0036v1
fatcat:3ftdvrgxcvdfzoatfmfu55roye
*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]

2010
*
arXiv
*
pre-print

This paper focuses on the

arXiv:1008.0377v1
fatcat:akefedjdp5elnm7j7lvubp5q2e
*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. ...##
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Intrinsic knotting and linking of almost complete graphs
[article]

2007
*
arXiv
*
pre-print

In addition, we classify

arXiv:math/0701422v1
fatcat:whz623khpjdwdnuyhgglkyxcum
*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]

2014
*
arXiv
*
pre-print

A

arXiv:1411.1837v1
fatcat:52ee3dskbzfphepomu4u6fuu3u
*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

2002
*
Journal of Graph Theory
*

We demonstrate four

doi:10.1002/jgt.10017
fatcat:zsqhi6vpazdfzcykyfvr2ak4jy
*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*[6] . 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]

2016
*
arXiv
*
pre-print

In particular, we highlight results related to

arXiv:1602.08122v2
fatcat:jciwfrni7bhblpktgb4jx7jpbq
*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*. [3], [67] No*intrinsically**knotted**graph*is 2-apex. ...##
###
Intrinsic linking and knotting in virtual spatial graphs

2007
*
Algebraic and Geometric Topology
*

We introduce a notion of

doi:10.2140/agt.2007.07.583
fatcat:pbq3jjm5mzhdfpy4riobpgycmu
*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]

2018
*
arXiv
*
pre-print

We present four models for a random

arXiv:1811.09726v1
fatcat:t24bgidujvah5izaoym5nhpqqy
*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*. ...

##
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Intrinsic linking and knotting in tournaments
[article]

2019
*
arXiv
*
pre-print

We consider

arXiv:1901.03451v1
fatcat:or6z57qfbbhfjf5eoomqnnvlca
*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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