Graphs With Disjoint Links in Every Spatial Embedding

Chan, S.C.; Dochtermann, Anton; Foisy, Joel; Hespen, Jennifer; Kunz, Eman; Lalonde, Trent L.; Loney, Quincy; Sharrow, Katherine; Thomas, Nathan

doi:10.1142/s0218216504003421

Cited by 7 publications

(26 citation statements)

References 3 publications

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“…Pair them to produce Z i . If either member of a pair has lk(Z (2) i , V 2 ) ≡ 0 mod 2, it advances to the next round. Otherwise, choose twelve paths connecting the members of each pair and produce A 1 .…”

Section: Multiple Component Constructionsmentioning

confidence: 99%

Intrinsically linked graphs and even linking number

Fleming

Diesl

2005

Algebr. Geom. Topol.

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We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A, L) = k2 r , k = 0, a non-split n-component link where all linking numbers are even, or an n-component link with components L, A i where lk(L, A i ) = 3k, k = 0. Links with other properties are considered as well.For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size of the complete graph is determined by the property in question.

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Section: Multiple Component Constructionsmentioning

confidence: 99%

Intrinsically linked graphs and even linking number

Fleming

Diesl

2005

Algebr. Geom. Topol.

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“…Every embedding of K 3,3,2 contains a pair of disjoint 3-cycles that have non-zero linking number [1]. As these 3-cycles are disjoint, they must be of the form xa i b j and ya k b l .…”

Section: Tournaments On 8 Verticesmentioning

confidence: 99%

“…We will construct intrinsically linked tournaments with the desired score sequences by iteratively assigning orientations to the edges of K 8 in a similar manner as the proof of Proposition 3. These edges form a subgraph of K 8 that is K 3,3,2 , and thus must contain a pair of 3-cycles L 1 , L 2 with non-zero linking number [1]. If these 3-cycles are both consistently oriented, we are done.…”

Section: Tournaments On 8 Verticesmentioning

confidence: 99%

Classifying Intrinsically Linked Tournaments by Score Sequence

Fleming¹,

Foisy²

2020

Preprint

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A tournament on 8 or more vertices may be intrinsically linked as a directed graph. We begin the classification of intrinsically linked tournaments by examining their score sequences. While many distinct tournaments may have the same score sequence, there exist score sequences S such that any tournament with score sequence S has an embedding with no nonsplit consistently oriented link. We call such score sequences linkless, and we show that the vast majority of score sequences for 8 vertex tournaments are linkless.We also extend these results to n vertex tournaments and are able to classify many longer score sequences as well. We show that for any n, there exist at least O(n) linkless score sequences, but we conjecture that the fraction of score sequences of length n that are linkless goes to 0 as n becomes large.

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“…K 9 has an embedding with no 3-link [6], so any 9 vertex tournament has an embedding with no non-split consistently oriented 3 component link. [2] that the graph D is intrinsically linked, and in every spatial embedding, there is a pair of linked cycles C 1 , C 2 with C 1 containing the edge d 1 d 2 . We similarly expand vertex c 1 in a second copy of K 3,3,2 with the same edge orientations and denote the resulting graph D .…”

Section: Intrinsic 3-linking In Tournamentsmentioning

confidence: 99%

“…Researchers have studied variations of these properties, such as requiring every embedding of the graph to contain cycles that form a non-split n-component link [4], a non-split link where one of more of the components are non-trivial knots [5] [8], or even more complex structures [12] [2].…”

Section: Introductionmentioning

confidence: 99%

Intrinsic linking and knotting in tournaments

Fleming

Foisy

2019

J. Knot Theory Ramifications

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A directed graph G is intrinsically linked if every embedding of that graph contains a non-split link L, where each component of L is a consistently oriented cycle in G. A tournament is a directed graph where each pair of vertices is connected by exactly one directed edge.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 ≤ n ≤ 154), intrinsically m-linked (3m ≤ n ≤ 8(2m − 3) 2 ), intrinsically linked with knotted components (9 ≤ n ≤ 107), and the disjoint linking property (12 ≤ n ≤ 14).We also introduce the consistency gap, which measures the difference in the order of a graph required for intrinsic n-linking in tournaments versus undirected graphs. We conjecture the consistency gap to be non-decreasing in n, and provide an upper bound at each n.

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Graphs With Disjoint Links in Every Spatial Embedding

Cited by 7 publications

References 3 publications

Intrinsically linked graphs and even linking number

Intrinsically linked graphs and even linking number

Classifying Intrinsically Linked Tournaments by Score Sequence

Intrinsic linking and knotting in tournaments

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