Intrinsic 3-linkedness is Not Preserved by Y∇ moves
IINTRINSIC 3-LINKEDNESS IS NOT PRESERVED BY Y ∇ MOVES
D. O’DONNOL
Abstract.
This paper introduces a number of new intrinsically 3-linked graphsthrough five new constructions. We then prove that intrinsic 3-linkedness isnot preserved by Y ∇ moves. We will see that the graph M , which is obtainedthrough a Y ∇ move on ( P G ) ∗∗ ( P G ), is not intrinsically 3-linked. Introduction
A graph, G , is intrinsically knotted if every embedding of G in R contains anontrivial knot. A link L is splittable if there is an embedding of a 2-sphere F in R (cid:114) L such that each component of R (cid:114) F contains at least one component of L .If L is not splittable it is called non-split . A graph, G , is intrinsically linked if everyembedding of G in R contains a non-split link. A graph, G , is minor minimal withrespect to being intrinsically linked (or simply minor minimal intrinsically linked )if G is intrinsically linked and no minor of G is intrinsically linked. The combinedwork of Conway and Gordon [3], Sachs [10], and Robertson, Seymour, and Thomas[8] fully characterizes intrinsically linked graphs. The graphs in the Petersen family,shown in Figure 1, are the complete set of minor minimal intrinsically linked graphs.So no minor of one of the graphs of the Petersen family is intrinsically linked, andevery graph that is intrinsically linked contains one of these graphs as a minor. Letthe set of the seven graphs of the Petersen family be denoted by PF .The concept of a graph being intrinsically linked can be generalized to a graphthat intrinsically contains a link of more than two components. A graph G is in-trinsically n -linked if every embedding of G in R contains a non-split n -componentlink. From here forward we will use n -link to mean a non-split n -component link.In this paper we focus on intrinsically 3-linked graphs. In Section 2, we discuss theset of known intrinsically 3-linked graphs and introduce our five new constructions.In Section 3, we prove that each of the new constructions results in an intrinsically3-linked graph.A Y ∇ move on an abstract graph is where a valance 3 vertex, v , together withits adjacent edges are deleted, and three edges are added, one between each pair ofvertices that had been adjacent to v . The reverse move is called a ∇ Y move . SeeFigure 6. In [10], Sachs showed that each graph in the Petersen family, i.e. all thosegraphs obtained from K by Y ∇ and ∇ Y moves, is also minor minimal intrinsicallylinked. Motwani, Raghunathan, and Saran [7] showed that both intrinsic linkednessand intrinsic knottedness are preserved by ∇ Y moves. Their proof that intrinsiclinkedness is preserved by ∇ Y moves immediately generalizes to show that intrinsic
Date : October 24, 2018.2010
Mathematics Subject Classification.
Primary 57M15, 57M25; Secondary 05C10.Supported in part by a NSF-AWM Mathematics Mentoring Travel Grant. a r X i v : . [ m a t h . G T ] O c t D. O’DONNOL K K , , G G G K − , G = P G
Figure 1.
This figure shows the graphs in the Petersen family,and the arrows indicate ∇ Y moves. Y ∇ move ∇ Y move
Figure 2.
The Y ∇ move and ∇ Y move. n -linkedness is also preserved by ∇ Y moves. Robertson, Seymour, and Thomas[8] showed that Y ∇ moves also preserve intrinsic linkedness. On the other hand,Flapan and Naimi [5] showed that Y ∇ moves do not preserve intrinsic knottedness.It is not known if intrinsic n -linkedness is preserved by Y ∇ moves in general. Thework in [8] showed that intrinsic 2-linkedness is preserved by Y ∇ moves. While thefamily of minor minimal intrinsically linked graphs (also minor minimal intrinsically2-linked graphs) is connected by Y ∇ and ∇ Y moves, the family of minor minimal
NTRINSIC 3-LINKEDNESS IS NOT PRESERVED BY Y ∇ MOVES 3 intrinsically 3-linked graphs is not [4]. It is also known that if the graph resultingfrom a Y ∇ move on a minor minimal intrinsically n -linked graph is intrinsically n -linked, then it is minor minimal intrinsically n -linked [2]. In Section 6, we provethat intrinsic 3-linkedness is not preserved by Y ∇ moves. Acknowledgements.
The author would like to thank Dorothy Buck, Erica Fla-pan, Kouki Taniyama and R. Sean Bowman for helpful conversations and theircontinued support. 2.
Intrinsically 3-linked graphs
There are a number of graphs already known to be intrinsically 3-linked. Figure3 shows all those graphs that have been shown to be intrinsically 3-linked, whereno minor is known to be intrinsically 3-linked (only G (2) is known to be minorminimal intrinsically 3-linked). In [6], Flapan, Naimi, and Pommersheim investigateintrinsically 3-linked graphs (or intrinsically triple linked graphs). They provedthat the complete graph on ten vertices, K is the smallest complete graph tobe intrinsically 3-linked. Bowlin and Foisy [1] also looked at intrinsically 3-linkedgraphs. They exhibited two different subgraphs of K that are also intrinsically3-linked, the graph resulting from removing two disjoint edges from K , call it K − { } , and the graph obtained by removing four edges incident to acommon vertex from K , call it K ∗ . So K is not minor minimal intrinsically3-linked. They also described two constructions that give intrinsically 3-linkedgraphs, the first being the graph that results from identifying an edge of G withan edge of G , when G and G are either K or K , ; we will call this graph G | G . Since all of the edges of K are equivalent as are those of K , this givesrise to three graphs. One of these graphs, K , | K , (also called J ), was previouslyshown to be intrinsically 3-linked in [4]. The second is the graph obtained byconnecting two graphs G , G ∈ PF by a 6-cycle where the vertices of the 6-cyclealternate between G and G . We will call such a graph ( G CG ) i . The subscript i is given because there can be multiple ways to combine the same two graphs inthis way. Not all of the vertices of each of the graphs of the Petersen family areequivalent, so there are many different ways to form the 6-cycle. Thus, there willbe many more than 7 + (cid:0) (cid:1) = 28 such graphs that one might first expect fromcombining the seven different graphs of PF in this construction. Due to the largenumber of graphs of this form they are not drawn in Figure 3 but instead a pictorialrepresentation of any such graph is shown. Flapan, Foisy, Naimi, and Pommersheimaddressed the question of minor minimal intrinsically n -linked graphs in [4], wherethey constructed a family of minor minimal intrinsically n -linked graphs. The minorminimal intrinsically 3-linked graph they constructed was called G (2), shown inFigure 3.In Section 3, we prove that the following five constructions give rise to intrinsi-cally 3-linked graphs: Let G , G ∈ PF and let v i be a vertex in the graph G i . Letthe set of adjacent vertices to the vertex v i be A i , for i = 1 , Construction 1 : Let the graph obtained by adding the edges between v andall but one of the vertices of A and the edges between v and all but one of thevertices of A to the graphs G and G be called ( G , v ) ∗∗ ( G , v ). We call this the double star construction . D. O’DONNOL G G K − { } K ∗ G (2) G CG K | K K | K , J = K , | K , Figure 3.
The set of of previously known intrinsically 3-linkedgraphs, for which no known minor is intrinsically 3-linked. (Thegraph G (2) is known to be minor minimal.) Construction 2 : Let the graph obtained by identifying the vertices v and v ,adding a vertex x and the edges from x to all but one of the vertices in the set A and all but one of the vertices in the set A be called ( G , v ) ∗ x ( G , v ). Construction 3 : The graph K − , has two vertices of valence three, label them x and y . Let G be one of the graphs of the Petersen family, let v be one of thevertices of G , and let A be the set of vertices adjacent to v in G . Let the graphobtained by identifying the two vertices x and v , and adding edges between y andall but one of the vertices in A be called K , ( G, v ). Construction 4 : Let the vertex identification construction be the constructionwhere a graph H is formed by adding edges between the sets of vertices A and A ,such that between every pair of vertices of A and pair of vertices of A there is atleast one edge joining a vertex from A to a vertex from A and then identifyingthe vertices v and v to get a single vertex x . Let the set of added edges between A and A be E n,m , where | A | = n and | A | = m . NTRINSIC 3-LINKEDNESS IS NOT PRESERVED BY Y ∇ MOVES 5
Construction 5 : Let V i be the full vertex set of G i . Let ( G ) ≡ = ( G ) be thegraph obtained by adding five disjoint edges between V and V to the graphs G and G .The vertex is dropped from the notation for Constructions 1, 2 and 3, if the ver-tices of the graph G or G are equivalent. These constructions introduce numerousnew intrinsically 3-linked graphs. For example, the graphs ( K ) ∗∗ ( K ), ( K ) ∗ x ( K )are both subgraphs of K | K . See Figure 4. So these two graphs together withall of the graphs that can be obtained from them by ∇ Y moves were previouslyunknown to be intrinsically 3-linked. Similarly, K , ( K ) is a subgraph of K | K , .So the graph K , ( K ) introduces a set of new intrinsically 3-linked graphs.3. New intrinsically 3-linked graphs
In this section we prove that the five new constructions explained in Section 2give intrinsically 3-linked graphs. These constructions exploit some nice propertiesof the graphs in the Petersen family.
Observation . For each G ∈ PF every pair of disjoint cycles contains all of thevertices of G .Thus every embedding of a graph from the Petersen family in R not only con-tains a 2-link but contains a 2-link which contains all of the vertices of the graph.It is known that, for any G ∈ PF , every embedding of G in R contains a twocomponent link with odd linking number [3, 10]. So we will work with linking mod(2) and denote the mod (2) linking number of two simple closed curves L and J by ω ( L, J ). (a) (b) (c) v v x x y Figure 4.
The graphs (a) ( K ) ∗∗ ( K ), (b) ( K ) ∗ x ( K ), and (c) K , ( K ).We will use the following lemma, proved in [1], to prove that Construction 1 ofthe previous section gives rise to an intrinsically 3-linked graph. Lemma 1.
In an embedded graph with mutually disjoint simple closed curves, C , C , C , and C , and two disjoint paths x and x , such that x and x begin in C and end in C , if ω ( C , C ) = ω ( C , C ) = 1 then the embedded graph contains anon-splittable 3-component link. D. O’DONNOL
Proposition 1.
Let ( G , v ) ∗∗ ( G , v ) be a graph obtained via Construction 1. Then ( G , v ) ∗∗ ( G , v ) is intrinsically 3-linked.Proof. Fix an arbitrary embedding of ( G , v ) ∗∗ ( G , v ). Since G is a graph in thePetersen family we know it must contain a 2-link and that v must be in one of thecomponents of the 2-link. Let the component that contains v be called C andthe other component be called C . Note that ω ( C , C ) = 1. Similarly, G mustcontain a 2-link and that v must be in one of the components of the 2-link. Letthe component that contains v be called C and the other component be called C , note that ω ( C , C ) = 1. Since v is in C two of the vertices adjacent to v arealso in C . At least one of these vertices must be adjacent to v , call it a . Note,the edge v a goes between C and C . Next, since v is in C two of the verticesadjacent to v are also in C and at least one of them is adjacent to v , call it b .The edge bv also goes between C and C . Notice that v a and bv cannot be thesame edges by construction. Thus by Lemma 1 we see that the chosen embeddingcontains a 3-link, and so ( G , v ) ∗∗ ( G , v ) is intrinsically 3-linked. (cid:3) To prove Proposition 2, we will use the following lemma which appears in [4]:
Lemma 2.
Suppose that G is a graph embedded in R and contains the simpleclosed curves C , C , C , and C . Suppose that C and C are disjoint from eachother and both are disjoint from C and C , and that C and C intersect in preciselyone vertex x . Also, suppose there are vertices u (cid:54) = x in C and v (cid:54) = x in C anda path P in G with endpoints u and v whose interior is disjoint from each C i . If ω ( C , C ) = ω ( C , C ) = 1 , then there is a non-splittable 3-component link in G . Proposition 2.
Let ( G , v ) ∗ x ( G , v ) be a graph obtained via Construction 2, then ( G , v ) ∗ x ( G , v ) is intrinsically 3-linked.Proof. Let A i be the sets vertices and x be the vertex as described in Construction2, in the previous section. Fix an arbitrary embedding of ( G , v ) ∗ x ( G , v ). Since G is in the Petersen family we know it must contain a 2-link and that v must be inone of the components of the 2-link. Let the component that contains v be called C and the other component be called C ; note that ω ( C , C ) = 1. Similarly, G must contain a 2-link and that v must be in one of the components of the 2-link.Let the component that contains v be called C and the other component be called C ; note that ω ( C , C ) = 1. Since v is in C there are two vertices in A that arealso in C and at least one of them is adjacent to x . Call it a i . Similarly, since v is in C there are two vertices of A that are also in C and at least one of themis adjacent to x . Call it b i . Let the path consisting of the two edges a i x and xb i be called P . The path P goes from C to C . So by Lemma 2 we see that theembedding contains a 3-link. Thus ( G , v ) ∗ x ( G , v ) is intrinsically 3-linked. (cid:3) The graph K , ( K ) is shown in Figure 4. We will use the following lemmasin the proof of the next proposition about Construction 3. See Section 2 for theconstructions. Lemma 3. [10]
Let K , be embedded in R , then every edge of K , is in a com-ponent of a 2-link. Lemma 4. [6]
Suppose that G is a graph embedded in R that contains the sim-ple closed curves C , C , C , and C . Suppose that C and C are disjoint from NTRINSIC 3-LINKEDNESS IS NOT PRESERVED BY Y ∇ MOVES 7 each other and both are disjoint from C and C , and that C ∩ C is an arc. If ω ( C , C ) = 1 and ω ( C , C ) = 1 , then there is a non-split 3-component link in G . Proposition 3.
Let K , ( G, v ) be a graph obtained via Construction 3, then K , ( G, v ) is intrinsically 3-linked.Proof. Let x , y , and A be as described in Construction 3. Fix an arbitrary embed-ding of K , ( G, v ). Since G is one of the graphs from the Petersen family it containsa 2-link C ∪ C which contains the vertex v = x , without loss of generality let x bein C . Since x is in C two of the vertices adjacent to x , are also in C . So at leastone of these vertices in C is adjacent to y , call the vertex a . Label the path P, thatis comprised of the two edges xa and ay . Notice xa ∈ C and ay / ∈ C ∪ C . Nowthe subgraph K − , together with the path P form a subdivision of K , , i.e. this canbe viewed as K , where P is one if the edges. By Lemma 3 for every embeddingof K , each edge is contained in a 2-link, so P is contained in a 2-link C ∪ C .Without loss of generality, let P be an edge of C . So C ∩ C = xa is an arc, ω ( C , C ) = 1 and ω ( C , C ) = 1. Thus by Lemma 4 the embedding of K , ( G, v )contains a 3-link. (cid:3) E E E E E E E E E E Figure 5.
Let | A | = n and | A | = m . This figure shows possiblesets of added edges E n,m for the vertex identification constructionsfor all different possible sizes of the vertex set A and A . Foreach of them the vertex sets A and A are shown vertically andadditional edges E n,m are shown.Recall Construction 4, the vertex identification construction, where a set of edges E n,m is added between the two sets of vertices A and A . For the full definitionrefer to Section 2. The graphs in the Petersen family have vertices of valence 3, D. O’DONNOL
4, 5, and 6. Let | A | = n and | A | = m . We want to construct E n,m , a set ofedges between A and A such that given any pair of vertices from A and anypair of vertices from A there is an edge between two of the vertices from thechosen pair. So each pair of vertices from A must be connected to m − A . To reduce the total number of edges needed we divide the edges evenlybetween the two vertices, so each vertex of A is connected to m − vertices of A .Suppose m ≥ n , this gives a lower bound of | E n,m | = ( m − ) n , if m is odd, and | E n,m | = ( m )( n −
1) + m − if m is even. Figure 5 shows possible sets of edges E n,m to be added between A and A for all possible combinations of valence. It canbe checked that there sets of vertices satisfy the criterion. However these are notthe only possible E n,m sets, and it is not known if they are optimal. In the caseof n = m = 3 then | E , | = 3 is the lower bound but in all other examples given | E n,m | is greater than the lower bound obtained. Proposition 4.
Any graph H constructed through the vertex identification con-struction of graphs G and G in the Petersen family is intrinsically 3-linked.Proof. Consider an arbitrary embedding of H . Let the notation be as in construc-tion 4: the identified vertex is labelled x , the sets of vertices adjacent to x in thesubgraphs G and G , respectively, are labelled A and A , and the set of edgesbetween A and A is E n,m . Since every vertex of a Petersen graph is contained inevery link in the embedding, x is in one of the components of the link in each G i .Let the components of the link in G be labelled C and C , with the vertex x inthe component C , and let the components of the link in G be labelled C and C ,with the vertex x in the component C . A pair of vertices from the set V is alsopart of C and, similarly, a pair of vertices from the set V is also part of C . Byconstruction, there is an edge, e , of the set E n,m between C and C . Since none ofthe edges of E n,m are contained in G or G , the interior of the edge e is disjointfrom the links C ∪ C and C ∪ C . Thus, by Lemma 2, H contains a 3-link. (cid:3) Proposition 5.
The graph ( G ) ≡ = ( G ) obtained by Construction 5, is intrinsically3-linked.Proof. Let V i be the vertex set of G i , let the set of five added edges be E . Fix anembedding of ( G ) ≡ = ( G ). Since G , G ∈ PF , G contains a 2-link that containsall of the vertices of V , call the link C ∪ C , and G contains a 2-link that containsall of the vertices of V , call the link C ∪ C . Because all of the vertices are inone of the 2-links each edge of E will go between components of the different 2-links. There are four different pairs of components that can be connected by thesaid edges, so by the pigeonhole principle two of the edges must go between thesame pair of components. By Lemma 1 the embedding contains a 3-link. Thus( G ) ≡ = ( G ) is intrinsically 3-linked. (cid:3) Intrinsic 3-linkedness is not preserved by Y ∇ moves In this section, we show that intrinsic 3-linkedness is not preserved by Y ∇ moves. Theorem 1.
Intrinsic 3-linkedness is not preserved by Y ∇ moves.Proof. We begin with (
P G ) ∗∗ ( P G ), since all of the vertices of
P G are equivalentthere is a single graph that can be obtained throughout the double star constructionwith two Petersen graphs. By Proposition 1, (
P G ) ∗∗ ( P G ) is intrinsically 3-linked.
NTRINSIC 3-LINKEDNESS IS NOT PRESERVED BY Y ∇ MOVES 9 ( P G ) ∗∗ ( P G ) M Figure 6.
The graph M is obtained from ( P G ) ∗∗ ( P G ) by a Y ∇ move on the indicated bold edges. a bc d ef g hj k Figure 7.
The spatial graph f ( M ). An embedding of M thatdoes not contain a 3-link.Let the graph obtained by a Y ∇ move on ( P G ) ∗∗ ( P G ) as indicated in Figure 6 becalled M .We claim that, the graph M is not intrinsically 3-linked. Consider the embedding f ( M ) shown in Figure 7. Let the vertices be labelled as indicated. Let K be theembedded subgraph defined by the vertices 1, 2, 3, 4, 5, 6, 7, 8, 9 and the edgesbetween them in f ( M ), and let K be the embedded subgraph defined by thevertices a, b, c, d, e, f, g, h, j, k and the edges between them. For f ( M ) to containa 3-link, the 3-link must be in both the embedded subgraphs K and K , sinceneither contains three disjoint simple closed curves on their own. Since K and K are disjoint and there is no linking between them, two of the edges joining thesubgraphs must also be in the 3-link. The subgraph K contains a single linked pair of cycles, indicated with thickened edges. Similarly, in K there is a single linkedpair of cycles, indicated with thickened edges. All other cycles in the subgraphs K and K bound disks that do not intersect the graph in their interiors. No pair ofedges between the two subgraphs K and K connects two of the linked cycles. Sothere is no 3-link in f ( M ). (cid:3) Notice that there are many graphs that can be constructed with the double starconstruction that are a Y ∇ move away from a graph that is not intrinsically 3-linked. Consider ( P G ) ∗∗ G for any G ∈ PF , a similar Y ∇ move to that is the proofabove will produce a graph that is not intrinsically 3-linked. More generally, thiscan be done with any double star construction where the vertex of A i that is notconnected to the vertex v j is trivalent. NTRINSIC 3-LINKEDNESS IS NOT PRESERVED BY Y ∇ MOVES 11
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