Some Ramsey-type results on intrinsic linking of n-complexes
aa r X i v : . [ m a t h . G T ] D ec SOME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES CHRISTOPHER TUFFLEY
Abstract.
Define the complete n -complex on N vertices , K nN , to be the n -skeleton of an( N − n -complexesin R n +1 necessarily exhibit complicated linking behaviour, thereby extending knownresults on embeddings of large complete graphs in R (the case n = 1) to higher dimen-sions. In particular, we prove the existence of links of the following types: r -componentlinks, with the linking pattern of a chain, necklace or keyring; 2-component links withlinking number at least λ in absolute value; and 2-component links with linking numbera non-zero multiple of a given integer q . For fixed n the number of vertices required foreach of our results grows at most polynomially with respect to the parameter r , λ or q . Introduction
In the 1980s Sachs [14] and Conway and Gordon [1] proved that an embedding of thecomplete graph K in R necessarily contains a pair of disjoint cycles that form a non-splitlink. This fact is expressed by saying that K is intrinsically linked . Conway and Gordonalso showed that every embedding of K in R contains a cycle that forms a nontrivialknot, and we say that K is intrinsically knotted .Since these papers, the study of intrinsic knotting and linking has been pursued inseveral directions, and we refer the reader to Ram´ırez Alfons´ın [13] for a survey of someknown results. One such direction is to show that embeddings of larger complete graphsnecessarily exhibit more complex knotting and linking behaviour. Restricting our atten-tion to linking, Flapan et al. [4] and Fleming and Diesl [6] have shown that embeddingsof sufficiently large complete graphs must contain non-split r -component links; Flapan [2]has shown that they must contain 2-component links with high linking number; and Flem-ing [5] has extended work by Fleming and Diesl [6] to show that, given an integer q , theymust contain 2-component links with linking number a nonzero multiple of q .We will refer to results such as those described above as Ramsey-type results on intrinsiclinking. Perhaps the strongest results in this direction are those of Negami [11] andFlapan, Mellor and Naimi [3]. Restricting attention to embeddings with a projection thatis a “good drawing”, Negami shows that, given a link L , for n, m sufficiently large everysuch embedding of the complete bipartite graph K n.m contains a link that is ambientisotopic to L . The restriction to embeddings with a projection that is a good drawingexcludes local knots in the edges, which is necessary but not sufficient (Negami [12]) forthe result to hold. With no restriction on the embedding, Flapan, Mellor and Naimi Date : June 12, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Intrinsic linking, complete n -complex, Ramsey Theory. n -COMPLEXES 2 show that intrinsic knotting and linking are arbitrarily complex in the following sense:Given positive integers r and α , embeddings of sufficiently large complete graphs contain r -component links in which the second co-efficient of the Conway polynomial of eachcomponent, and the linking number of each pair of components, is at least α in absolutevalue.Extending the result of Sachs [14] and Conway and Gordon [1] in another direction, wemay consider embeddings of n -complexes in R d . By a general position argument every n -complex embeds in R n +1 , and a pair of disjoint n -spheres in R n +1 have a well definedlinking number (the homology class of one component in the n th homology group of thecomplement of the second, which is isomorphic to Z ), so we take d = 2 n + 1. Define the complete n -complex on N vertices , K nN , to be the n -skeleton of an N − K n n +4 is intrinsically linked, in the sense thatevery embedding in R n +1 contains a pair of disjoint n -spheres that have nonzero linkingnumber. Since K N ∼ = K N this specialises to the K result in the case n = 1.The purpose of this paper is to establish some Ramsey-type results for embeddings ofcomplete n -complexes in R n +1 . Our results are already known for embeddings of completegraphs in R , and our arguments will typically mimic the proof of the corresponding 1-dimensional result. However, in the case of Theorem 1.4 we will obtain a better boundfor n = 1 than that previously known; and in addition, some constructions used in thearguments require modifications in higher dimensions. These modifications are neededfor two main reasons: Firstly, ∂D n = S n − is disconnected for n = 1, but not for n ≥ D n have simpler combinatorics for n = 1 than they do for n ≥ n ≥ n -sphere does not knot in R n +1 for reasons of co-dimension,and an arbitrary n -complex does not necessarily embed in R n +2 . Thus, we will not seekto establish any results on intrinsic knotting of complete n -complexes.1.1. Statement of results.
In what follows, a k -component link means k disjoint n -spheres embedded in R n +1 . Given a 2-component link L ∪ L we will write ℓk ( L , L ) fortheir linking number, and ℓk ( L , L ) for their linking number mod two. For { i, j } = { , } the integral linking number is given by the homology class [ L i ] in H n ( R n +1 − L j ; Z ) ∼ = Z .Our first result is similar to Theorems 1 and 2 of Flapan et al. [4], and shows thatembeddings of sufficiently large complete n -complexes necessarily contain non-split r -component links. Moreover, the number of vertices required grows at most linearly withrespect to each of r and n . Theorem 1.1.
Let r ∈ N , r ≥ . (a) For N ≥ (2 n + 4)( r − every embedding of K nN in R n +1 contains an r -componentlink L ∪ L ∪ · · · ∪ L r such that ℓk ( L i , L i +1 ) = 0 (1.1) for i = 1 , . . . , r − . (b) If r ≥ then for N ≥ (2 n + 4) r every embedding of K nN in R n +1 contains an r -component link L ∪ L ∪ · · · ∪ L r satisfying equation (1.1) for i = 1 , . . . , r (subscripts taken mod r ). OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 3 The link of Theorem 1.1(a) resembles a chain, and the link of Theorem 1.1(b) resemblesa necklace, except that there is no requirement that non-adjacent components do not alsolink. Our next result generalises Lemma 2.2 of Fleming and Diesl [6], and yields linksthat resemble a bunch of keys on a keyring. However, there is again no requirement thatthe “keys” do not also link each other, and following Flapan et al. [3] we call such a linka generalised keyring . Generalised keyrings will play a crucial role in establishing ourresults for 2-component links, in Theorems 1.3–1.5.
Theorem 1.2.
For a natural number r define κ n ( r ) = 4 r (2 n + 4) + n + (cid:24) r − n (cid:25) + 1 . Then every embedding of K nκ n ( r ) in R n +1 contains an ( r + 1) -component link R ∪ L ∪ L ∪ · · · ∪ L r such that ℓk ( R, L i ) = 1 for i = 1 , . . . , r . Observe that κ n ( r ) grows quadratically in r and linearly in n . The existence of gener-alised keyrings in embeddings of K nN for N sufficiently large may be established by follow-ing Fleming and Diesl’s argument, or that of Flapan et al. [3, Lemma 1]; the Fleming-Dieslargument leads to a bound that grows exponentially with respect to r , and so we willfollow the argument of Flapan et al., as this leads to the polynomial bound given above.For n = 1 the term n + ⌈ (4 r − /n ⌉ + 1 of κ n is not needed, so it suffices to take κ ( r ) = 24 r . This bound follows from Flapan et al. [3, Lemma 1], although they do notstate the bound explicitly.Our last three results concern linking number in 2-component links. The first extendsTheorem 2 of Flapan [2] to higher dimensions (although our proof will be based on atechnique from Lemma 2 of Flapan et al. [3], as this leads to a better bound in higherdimensions): Theorem 1.3.
Let λ ∈ N be given, and let N = κ n (2 λ −
1) + n + (cid:24) λ − n (cid:25) + 1 . Then every embedding of K nN in R n +1 contains a two-component link L ∪ J such that, forsome orientation of the components, ℓk ( L, J ) ≥ λ . Our last two results concern divisibility of the linking number. Fleming and Diesl [6]showed that for q = 3 or q a power of two, embeddings of sufficiently large complete graphsin R necessarily contain 2-component links with linking number a nonzero multiple of q ,and Fleming [5] later extended this to all q ∈ N . We now extend this further to embeddingsof complete n -complexes in R n +1 , and by slightly modifying Fleming’s argument, reducethe number of vertices required from exponentially many to only polynomially many. Westate and prove two results in this direction: the first is for q arbitrary, and the second isfor q = p prime, where a simpler argument leads to a bound with much slower growth. OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 4 Theorem 1.4.
Let q be a positive integer. Then for N sufficiently large every embeddingof K nN in R n +1 contains a two-component link R ∪ S such that ℓk ( R, S ) = kq for some k = 0 . The number of vertices required grows no faster than c ( n + 1) (cid:0) n +4 n +1 (cid:1) q n +2 ( c aconstant), which for fixed n grows polynomially in q . When q = p is prime, a much simpler argument leads to a bound with growth O ( p )instead of O ( q n +2 ): Theorem 1.5.
Let p ∈ N be prime, and let N = κ n (2 p −
1) + n + (cid:24) p − n (cid:25) + 1 . Then every embedding of K nN in R n +1 contains a two-component link L ∪ J such that ℓk ( L, J ) = kp for some k = 0 . Since the proof of Theorem 1.5 is simpler than that of Theorem 1.4 we will prove itfirst, in Section 4.2, and then prove Theorem 1.4 later in Section 6.For n = 1, Theorem 1.4 may be proved using a total of4 q (6 + 15( q − q (5 q − q vertices rather than O ( q log q ). In fact itshould be possible to reduce the number of vertices required further, by a factor of about2 /
3, because for n = 1 our method really only requires the keys to have about 2 q vertices.For large n Stirling’s formula may be used to show that asymptotically we have c ( n + 1) (cid:18) n + 4 n + 1 (cid:19) q n +2 ∼ C √ n n q n +2 . Thus the number of vertices required grows at most exponentially with respect to n .1.2. Discussion.
We briefly discuss the existence of more complex links in embeddingsof large complete complexes in R n +1 .1.2.1. More complex keyrings.
Each of Theorems 1.3–1.5 is proved by converting a suitablegeneralised keyring R ∪ L ∪ · · · ∪ L m into a two component link R ∪ L ′ , where L ′ is formedas a connect sum of some of the L i (and perhaps an additional disjoint component S ).Starting with a generalised keyring with mr keys, and working with them m at a time,we may therefore construct a link R ∪ L ′ ∪ · · · ∪ L ′ r in which each linking number ℓk ( R, L ′ i )satisfies the conclusion of the theorem. It follows for example that for q ∈ N and N sufficiently large, every embedding of K nN in R n +1 contains a link R ∪ L ′ ∪ · · · ∪ L ′ r inwhich each linking number ℓk ( R, L ′ i ) is a nonzero multiple of q . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 5 More complex linking patterns.
Flapan et al. [3, Theorem 1] show that intrinsiclinking of graphs in R is arbitrarily complex in the following sense: Given natural numbers r and λ , for N sufficiently large every embedding of K N in R contains an r -componentlink in which all pairwise linking numbers are at least λ in absolute value. We believethat, with minor adaptions to higher dimensions, their work shows that intrinsic linking of n -complexes in R n +1 is arbitrarily complex in this sense also. The main adaption neededis to use our Lemma 2.5 in place of the 1-dimensional construction it replaces in higherdimensional arguments. This adaption requires the addition of some extra vertices (tocreate the auxiliary sphere S of the lemma), and is illustrated in the proofs of Lemma 3.2and Theorem 1.3. These are based respectively on their Lemma 1 and a technique fromthe proof of their Lemma 2.A step in their argument is to show that, for N sufficiently large, every embedding of K N in R contains a link X ∪ · · · ∪ X m ∪ Z ∪ · · · ∪ Z m such that ℓk ( X i , Z j ) = 1for 1 ≤ i, j ≤ m (Flapan et al. [3, Prop. 1]). We observe that this step certainly extendsto embeddings of complete n -complexes in R n +1 , as their proof is a purely combinatorialargument that depends only on their Lemma 1 and the existence of generalised keyrings,which we extend here to higher dimensions as Lemma 3.2 and Theorem 1.2 respectively.1.3. Organisation.
The paper is organised as follows. We begin with some technical pre-liminaries in Section 2, and then prove Theorems 1.1 and 1.2 concerning many-componentlinks in Section 3. In Section 4 we prove our first two results on linking numbers in 2-component links, Theorems 1.3 and 1.5.We then construct some triangulations of an M -simplex in Section 5, as further technicalpreliminaries needed for our proof of our divisibility result Theorem 1.4. This result isproved in Section 6. As a further application of the triangulations of Section 5 we concludethe paper in Section 7 with an alternate proof of Theorem 1.3, without the polynomialbound on the number of vertices required. This introduces an additional technique thatmay be used to prove Ramsey-type results on intrinsic linking of n -complexes.2. Technical preliminaries I: Spheres and discs in K nN In this section we construct some subcomplexes of K nN that are needed for our proofs.As an aid to understanding, in Section 2.1 we first illustrate the role the correspondingsubcomplexes of K N play in studying intrinsic linking of graphs in R .2.1. Tactics.
A common technique of [2, 3, 4, 5, 6] in proving Ramsey-type results forgraphs is the use of connect sums and the additivity of linking number. These may beused to convert a link with several components to one with fewer components, but morecomplicated linking behaviour. We illustrate this technique by sketching the proofs for n = 1 of the four-to-three Lemmas 3.1 and 7.2. The n = 1 case of Lemma 7.2 correspondsto Lemma 2 of Flapan [2], and Lemma 3.1 is a mod two version of this result that issimilar to Lemma 1 of Flapan et al. [4].Suppose that the 4-component link Y ∪ X ∪ X ∪ Y in Figure 1(a) is part of anembedding of K N in R , and that we wish to replace the cycles X and X with a singlecycle X linking both Y and Y mod two. We choose vertices v , v on X and w , w on OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 6 PSfrag replacements (a) (b) (c) (d) XX X Y Y Y Y v v w w Figure 1.
Illustrating the proof of Lemma 3.1 (the four-to-three lemmafor mod two linking number) in the case n = 1. X , and consider the edges ( v i , w i ) as in Figure 1(b). Together with X and X these giveus a collection of cycles (Figure 1(c)) whose linking numbers with each of Y and Y sumto zero mod two; and taking the connect sum of a suitably chosen subset as in Figure 1(d)we get the desired cycle X .Working now with integer co-efficients, consider the link Y ∪ X ∪ X ∪ Y in Figure 2(a).Our goal here is to replace this with a three component link L ∪ Z ∪ W such that ℓk ( L, Z )is nonzero, and ℓk ( L, W ) is at least as large as ℓk ( X , Y ) in absolute value. We againdo this by constructing a series of cycles that sum to zero with X and X , but now inorder to ensure we can find one linking Y with the correct sign it is necessary to haveat least q > | ℓk ( X , Y ) | such cycles. This is achieved by choosing vertices v , . . . , v q on X and w , . . . , w q on X , such v , . . . , v q are encountered in increasing order followingthe orientation of X , and w , . . . , w q are encountered in decreasing order following theorientation of X . The needed cycles are formed by connecting X and X using theedges ( v , w ) , . . . , ( v q , w q ), as in Figure 2(b), and a suitable connect sum (Figure 2(c))then gives us the desired 3-component link.To prove analogous results in higher dimensional dimensions we will regard the intervals[ v , v q ] and [ w , w q ] as identically triangulated discs D ⊆ X and D ⊆ X , and thecorrespondence v i w i as an orientation reversing simplicial isomorphism φ : D → D mapping one triangulation to the other. Given this data we then construct the collectionof edges ( v i , w i ), which we regard as a complex C homeomorphic to D (0)1 × I realising therestriction of φ to the zero skeleton of D . The pair of edges ( v i , w i ) and ( v i +1 , w i +1 ) maythen be seen as a copy of S × I , which we cap with the intervals [ v i , v i +1 ] , [ w i , w i +1 ] tocreate a copy of S .Triangulations of an interval have very simple combinatorics, and in Figure 2(b) itdidn’t matter that there was an additional vertex between w and w . Thus, Flapan’s OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 7 PSfrag replacements (a) (b) (c) XX X Y Y Y Y v v v v w w w w Figure 2.
Illustrating the proof of Flapan’s Lemma 2, the n = 1 case ofour Lemma 7.2 (the four-to-three lemma for integral linking number).argument only requires that each component has at least q vertices. In order to usesimilar techniques when n ≥ D n . Additional work will thenbe required to ensure that our links contain such discs.2.2. Cylinders, spheres and discs in K nN . We now construct the needed subcomplexesof K nN . Lemma 2.1.
Let ( S , D ) and ( S , D ) be disjoint subcomplexes of K nN each homeomor-phic to ( S n , D n ) . Suppose that there is a simplicial isomorphism φ : D → D . Let D ( n − i be the ( n − -skeleton of D i . Then there is a subcomplex C of K nN and ahomeomorphism Φ : D ( n − × I → C such that (1) all vertices of C lie on D ∪ D ; (2) C ∩ S i = D ( n − i for i = 1 , ; (3) Φ restricts to the identity on D ( n − × { } ; and (4) Φ = φ on D ( n − × { } . We note that the subcomplex C may be regarded as the mapping cylinder of the re-striction of φ to the ( n − Proof.
To construct C we use the subdivision of ∆ k × I into ( k + 1)-simplices used in theproof of the homotopy invariance of singular homology (see for example Hatcher [7, p.112]). Label the vertices of D arbitrarily as v , v , . . . , v M , and label the vertices of D as w , w , . . . , w M so that w i = φ ( v i ). Now, for each k -simplex δ = [ v i , . . . , v i k ] of D ( n − ,with i < i < · · · < i k , we have δ × I ∼ = C ( δ ) = k [ j =0 [ v i , . . . , v i j , w i j , . . . , w i k ] . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 8 Since k ≤ n − k + 1) simplex involved in this union is a simplex of K nN , and weobtain a subcomplex of K nN homeomorphic to δ × I , meeting D and D in δ × { } = δ and δ × { } = φ ( δ ) respectively. In addition, all vertices of C ( δ ) belong to D ∪ D .Let δ l denote the simplex [ v i , . . . , ˆ v i l , . . . , v i k ] belonging to ∂δ , where the hat indicatesthat v i l is omitted. A k -simplex belonging to C ( δ ) is of one of several possible types:(1) the simplex [ v i , . . . , v i k ] = δ or [ w i , . . . , w i k ] = φ ( δ );(2) one of the simplices [ v i , . . . , ˆ v i l , . . . , v i j , w i j , . . . , w i k ]or [ v i , . . . , v i j , w i j , . . . , ˆ w i l , . . . , w i k ]with l fixed and j = l , which together make up C ( δ l );(3) a simplex of the form [ v i , . . . , v i j , w i j +1 , . . . , w i k ], which is interior to δ × I .Inductively, this implies that if δ ′ is a simplex of δ , then C ( δ ′ ) is a subcomplex of C ( δ ),and the diagram δ ′ × I −−−→ δ × I ∼ = y y ∼ = C ( δ ′ ) −−−→ C ( δ )commutes. Moreover, our construction ensures that C ( δ ) and C ( δ ) are disjoint unless δ and δ intersect, in which case C ( δ ) ∩ C ( δ ) = C ( δ ∩ δ ). Thus, taking the union of C ( δ )over all ( n − D ( n − we obtain a subcomplex C of K nN homeomorphic to D ( n − × I meeting S i in D ( n − i for each i , and the homeomorphism Φ may be constructedsatisfying the given conditions. (cid:3) Corollary 2.2.
Let ( S , D ) and ( S , D ) be disjoint subcomplexes of K nN each homeomor-phic to ( S n , D n ) . Suppose that there is an orientation reversing simplicial isomorphism φ : D → D , and let ∆ , . . . , ∆ k be the n -simplices of D . Then there are subcomplexes P , P , . . . , P k of K nN such that (1) the vertices of P , P , . . . , P k all lie on S ∪ S ; (2) P i ∼ = S n for each i ; (3) P ∩ S j = S j \ D j for i = 1 , ; (4) P i ∩ S = ∆ i , P i ∩ S = φ (∆ i ) for i ≥ ; and (5) as an integral chain we have S + S + k X i =0 P i = 0 . Remark . Condition (1) implies that if A is a subcomplex of K nN disjoint from S ∪ S ,then A is disjoint from P i for all i . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 9 Proof.
We obtain the required spheres P i using the subcomplex C and homeomorphismΦ : D ( n − × I → C constructed in Lemma 2.1 above. For each i = 1 , . . . , k let P i = ∆ i ∪ φ (∆ i ) ∪ Φ( ∂ ∆ i × I ) , and let P = S \ D ∪ S \ D ∪ Φ( ∂D × I ) . Then Lemma 2.1 ensures that each P i is a subcomplex of K nN satisfying conditions (1)–(4)above.To obtain (5) we must orient each sphere P i . For i ≥ P i so that ∆ i receives the opposite orientation from P i as it does from S , and we orient P analogouslyusing the disc S \ D . This ensures that φ ♯ ∆ i receives opposite orientations from S and P i also, since φ is orientation reversing on ∆ i with respect to both S and P i (on P i ∼ = S n it is induced by reflection in an equatorial S n − ). Similar considerations applyto P , as φ extends to a (not necessarily simplicial) orientation reversing homeomorphism( S , S \ D ) → ( S , S \ D ).It remains to consider the subcomplexes C ( δ ), for δ an ( n − D . Eachsuch simplex belongs to two n -simplices of S , and receives opposite orientations fromeach (since ∂S = 0); consequently, each subcomplex C ( δ ) belongs to two spheres P i and P j , and is also oppositely oriented by each. This completes the proof. (cid:3) Remark . The n -spheres P i of Corollary 2.2 may be expressed explicitly as chains asfollows. We assume throughout that all simplices of D are written with the labels ontheir vertices in increasing order.For each k -simplex δ = [ v i , . . . , v i k ] of D ( n − define P ( δ ) = k X j =0 ( − j [ v i , . . . , v i j , w i j , . . . , w i k ] . Let ε i ∈ {± } be the co-efficient of ∆ i in the chain S , and set P i = − ε i (∆ i + P ( ∂ ∆ i ) − φ ♯ (∆ i ))for i ≤
1, and P = ( D − S ) + ( D − S ) + P ∂D . We verify below that ∂P i = 0, and that S + S + P i P i = 0.Suitably adapted, the calculation on page 112 of Hatcher [7] shows that ∂ P = φ ♯ − id ♯ −P ∂, so for i ≥ − ε i ∂P i = ∂ ∆ i + ∂ P ∂ ∆ i − ∂φ ♯ ∆ i = ∂ ∆ i + φ ♯ ∂ ∆ i − id ♯ ∂ ∆ i − P ∂ ∆ i − φ ♯ ∂ ∆ i = 0 . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 10 PSfrag replacements S S S S S S S S δ δ δ Figure 3.
Illustrating Lemma 2.5 in the case n = 1, k = 3. The sphere S is used to construct a sphere S meeting S i in a single n -simplex δ i for i = 1 , , ∂P = ∂ ( D − S ) + ∂ ( D − S ) + ∂ P ∂D = ∂D + ∂D + φ ♯ ∂D − id ♯ ∂D − P ∂ D = ∂D + ∂D − ∂D − ∂D ( φ ♯ ∂D = ∂φ ♯ D = − ∂D )= 0 , as required. Summing, we have D = P ki =1 ε i ∆ i , so k X i =1 P k = − k X i =1 ε i ∆ i − P ∂ k X i =1 ε i ∆ i + φ ♯ k X i =1 ε i ∆ i = − D − P ∂D + φ ♯ D = − D − P ∂D − D = − P − S − S , and it follows that S + S + P ki =0 P i = 0.2.3. Connect sums of several spheres.
Our next technical lemma takes several spheres S , . . . , S k , and an additional sphere S , and constructs a sphere S meeting each of S , . . . , S k in a single n -simplex. The case n = 1, k = 3 is illustrated in Figure 3. Thislemma is an adaption to higher dimensions of a construction used by Flapan et al. [3]in the case n = 1. In that case the additional sphere S is not needed, as it is onlynecessary to choose edges joining S i to S i +1 , and S k to S . This depends on the fact thatthe cylinder S × I is disconnected, and our additional sphere S is necessary for n ≥ S n − × I is connected. Lemma 2.5.
Let S , S , . . . , S k be disjoint subcomplexes of K nN each homeomorphic to S n , and suppose that S has at least k n -simplices. Then there is a subcomplex S of K nN such that (1) the vertices of S all lie on S ∪ · · · ∪ S k ; (2) S is homeomorphic to S n ; (3) for i = 1 , . . . , k there is an n -simplex δ i of S i such that S ∩ S i = δ i . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 11 Moreover, if each sphere S i is oriented, then S may be chosen and oriented such that δ i receives opposite orientations from S and from S i .Proof. We will assume that the S i are oriented. Choose an n -simplex δ i belonging to S i for each i ≥
1, distinct n -simplices δ ′ i belonging to S for i = 1 , . . . , k , and orientationreversing simplicial isomorphisms φ i : δ i → δ ′ i . Applying Corollary 2.2 to the pairs ( S i , δ i )and ( S , δ ′ i ) we obtain a sphere Q i with all its vertices on S i ∪ S , and such that Q i meets S i in δ i and S in δ ′ i . Note that this implies Q i ∩ Q j = δ ′ i ∩ δ ′ j .We set T = S , and for i = 1 , . . . , k we inductively define T i to be the complex obtainedfrom T i − and Q i by omitting the interior of the disc δ ′ i . Then at each stage T i is an n -sphere, because it is the result of gluing two discs along their common boundary ∂δ ′ i , andsetting S = T k we obtain the desired subcomplex. (cid:3) To conclude this section we establish a bound on the number of vertices required toconstruct an n -sphere with a specified number of n -simplices. Lemma 2.6.
Given ℓ ∈ N there is a triangulation of S n with n + ℓ + 1 vertices and ℓn + 2 n -simplices.Proof. We construct the triangulation from a suitable triangulation of D n +1 with ℓ ( n +1)-simplices. For i = 1 , . . . , ℓ let ∆ i be an ( n + 1)-simplex, and choose distinct n -simplices δ i , σ i belonging to ∆ i . Choose a simplicial isomorphism φ i : δ i → σ i +1 for each i = 1 , . . . , ℓ − D be the ( n + 1)-disc that results from gluing the ∆ i according to the φ i . Weclaim that S = ∂D is the required triangulated n -sphere.The union ∆ ∪· · ·∪ ∆ ℓ has a total of ℓ ( n +2) n -simplices, of which 2( ℓ −
1) are identifiedin pairs to form D . The n -simplices involved in the identifications lie in the interior of D , and the rest on the boundary, so S has ℓ ( n + 2) − ℓ −
1) = ℓn + 2 n -simplices,as claimed. Similarly, each gluing identifies 2( n + 1) vertices in pairs, leaving a total of ℓ ( n + 2) − ( n + 1)( ℓ −
1) = ℓ + n + 1; alternately, we may carry the gluings out sequentially,and we see that we start with n + 2 vertices, and each gluing adds just one, for a total of( n + 2) + ( ℓ −
1) = n + ℓ + 1.To complete the proof we show that the vertices of D all lie on S . For n = 1 a circlewith ℓ + 2 edges necessarily has ℓ + 2 vertices, by Euler characteristic; while for n ≥ i belongs to at least three n -simplices, and so to at least one n -simplexbelonging to ∂D after the identifications. (cid:3) Corollary 2.7. If k ∈ N and N ≥ n + ⌈ k/n ⌉ + 1 then K nN contains a subcomplex S ∼ = S n with at least k + 2 n -simplices.Proof. Set ℓ = ⌈ k/n ⌉ . Then ℓ ∈ N and ℓ ≥ k/n , so the construction of Lemma 2.6 yieldsan n -sphere S in K nN with at least k + 2 n -simplices. (cid:3) Many-component links
We now prove Theorems 1.1 and 1.2, thereby showing that embeddings of sufficientlylarge complete complexes necessarily contain non-split links with many components.
OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 12 Necklaces and chains.
In this section we establish Theorem 1.1. The key step isthe following lemma, which plays the role of Lemma 1 in Flapan et al. [4].
Lemma 3.1 (The four-to-three lemma for mod two linking number) . Let Y ∪ X ∪ X ∪ Y be a -component link contained in some embedding of K nN in R n +1 , satisfying ℓk ( X , Y ) = ℓk ( X , Y ) = 1 . Then there is an n -sphere X in K nN , all of whose vertices lie on X ∪ X , such that ℓk ( Y , X ) = ℓk ( X, Y ) = 1 . Proof. If ℓk ( X , Y ) = 1 then we may simply let X = X , and if ℓk ( X , Y ) = 1 then wemay simply let X = X . So suppose that ℓk ( X , Y ) = ℓk ( X , Y ) = 0 . Choose n -simplices δ , δ belonging to X , X respectively, and apply Corollary 2.2 to thepairs ( X , δ ), ( X , δ ) to obtain spheres P , P satisfying X + X + P + P = 0 . In the homology groups H n ( R n +1 − Y i ; Z / Z ) we have[ X ] + [ X ] + [ P ] + [ P ] = 0 , and since [ X ] + [ X ] = 1 in each group we have also [ P ] + [ P ] = 1 in each group. Hence,for each i , precisely one of [ P ], [ P ] must equal 1 in H n ( R n +1 − Y i ; Z / Z ).If [ P ] takes the same value in both groups then we are done by setting X = P if[ P ] = 0 in both groups, and X = P if [ P ] = 1. Otherwise, without loss of generalitysuppose that [ P ] is zero in H n ( R n +1 − Y ; Z / Z ) and nonzero in H n ( R n +1 − Y ; Z / Z ),and let X be the n -sphere obtained from X and P by omitting the interior of the simplex δ . Then [ X ] = [ X ] + [ P ] = ( [ X ] = 1 in H n ( R n +1 − Y ; Z / Z ) , [ P ] = 1 in H n ( R n +1 − Y ; Z / Z ) , and the result follows. (cid:3) We now prove Theorem 1.1, using the above lemma.
Proof of Theorem 1.1.
The proof of part (a) is by induction on r , with the base case r = 2 given by Taniyama [15], and the inductive step following from Lemma 3.1. Given anembedding of K n (2 n +4) r in R n +1 , choose disjoint copies of K n (2 n +4)( r − and K n n +4 containedin the embedding. By the inductive hypothesis the K n (2 n +4)( r − contains an r -componentlink L ∪ L ∪· · ·∪ L r satisfying equation (1.1) for i = 1 , . . . , r −
1, and the K n n +4 contains atwo component link J ∪ K such that ℓk ( J, K ) = 1. Applying Lemma 3.1 to the (ordered)link L r − ∪ L r ∪ J ∪ K we obtain an n -sphere X with all its vertices on L r ∪ J such that ℓk ( L r − , X ) = ℓk ( X, K ) = 1 . The link L ∪ · · · ∪ L r − ∪ X ∪ K is then the desired r -component link.To prove (b) we apply Lemma 3.1 to suitably chosen components of an ( r +1)-componentlink as given by part (a). Given an embedding of K n (2 n +4) r in R n +1 , there is an ( r + 1)-component link L ∪ L ∪ · · · ∪ L r ∪ L r +1 satisfying equation (1.1) for i = 1 , . . . , r . We OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 13 apply Lemma 3.1 to the (ordered) link L r ∪ L r +1 ∪ L ∪ L to obtain an n -sphere X , withall its vertices on L r +1 ∪ L , and satisfying ℓk ( L r , X ) = ℓk ( X, L ) = 1 . The link L ∪ · · · ∪ L r ∪ X is then the desired r -component link. (cid:3) Generalised keyrings.
We prove Theorem 1.2, by extending Lemma 1 of Fla-pan et al. [3] to higher dimensions in the following form.
Lemma 3.2.
Let K nN be embedded in R n +1 such that it contains a link L ∪ J ∪ · · · ∪ J m ∪ X ∪ · · · ∪ X m , where L has at least m n -simplices, and ℓk ( J i , X i ) = 1 for all i . Then there is an n -sphere Z in K nN with all its vertices on L ∪ J ∪ · · · ∪ J m , and an index set I with | I | ≥ m ,such that ℓk ( Z, X j ) = 1 for all j ∈ I .Proof. The argument is that of Flapan et al. [3], with the addition of the component L needed to create the analogue of their cycle C connecting the J i .Since L has at least m simplices we may apply Lemma 2.5 to the (ordered) link L ∪ J ∪ · · · ∪ J m , obtaining an n -sphere S with all its vertices on L ∪ J ∪ · · · ∪ J m andmeeting each sphere J i in an n -simplex δ i . If at least m of the mod two linking numbers ℓk ( S , X i ) are nonzero then we are done by setting Z = S , so we assume in what followsthat fewer than m of these mod two linking numbers are nonzero.Following Flapan et al. we define M to be the m × m matrix over Z / Z with ij -entry M ij = ℓk ( J i , X j ). Let r i be the i th row of M . Then M ii = 1 for all i , and Flapan et al.use this to show that there are indices i , . . . , i k such that V = r i + · · · + r i k has at least m entries that are equal to 1. Let Z be the n -sphere obtained from S and J i , . . . , J i k by omitting the interiors of the simplices δ i , . . . , δ i k . We claim that Z is therequired n -sphere.Indeed, for j = 1 , . . . , m we have ℓk ( Z, X j ) = ℓk ( S , X j ) + k X ℓ =1 ℓk ( J i ℓ , X j ) = ℓk ( S , X j ) + V j , (3.1)where V j = P kℓ =1 ℓk ( J i ℓ , X j ) is the j th entry of V . By construction at least m of the V j are nonzero, and by assumption fewer than m of the ℓk ( S , X j ) are nonzero. Hence thereare at least m − m = m indices j for which V j = 1 while ℓk ( S , X j ) = 0. Consequently,the set I = { ≤ j ≤ m : ℓk ( S , X j ) = V j } has at least m elements. But ℓk ( Z, X j ) = 1if and only if j ∈ I , by (3.1), so we are done. (cid:3) We now obtain Theorem 1.2 as a corollary to Lemma 3.2 and Corollary 2.7.
Proof of Theorem 1.2.
Recall that κ n ( r ) = 4 r (2 n + 4) + n + (cid:24) r − n (cid:25) + 1 , OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 14 and for ease of notation let ℓ = ⌈ (4 r − /n ⌉ . Given an embedding of K nκ n ( r ) in R n +1 ,choose 4 r disjoint copies of K n n +4 contained in the embedding, together with a copy of K nn + ℓ +1 . By Taniyama [15] the i th copy of K n n +4 contains a 2-component link J i ∪ X i suchthat ℓk ( J i , X i ) = 1, and by Corollary 2.7 the copy of K nn + ℓ +1 contains an n -sphere L withat least 4 r n -simplices. The result now follows by applying Lemma 3.2 with m = 2 r tothe link L ∪ J ∪ · · · ∪ J r ∪ X ∪ · · · ∪ X r . (cid:3) Linking number in 2-component links
We now prove Theorems 1.3 and 1.5, concerning the linking number in a 2-componentlink. To prove each result we start with a suitable generalised keyring, and combine someof the “keys” to obtain the second component of the desired link.4.1.
Bounding the absolute value of the linking number from below.
Proof of Theorem 1.3.
We use a technique of Flapan et al. [3] from the proof of theirLemma 2. For simplicity of notation let ℓ = ⌈ (2 λ − /n ⌉ , and choose disjoint copies of K nκ n (2 λ − and K nn + ℓ +1 contained in K nN . Given an embedding of K nN in R n +1 , the copyof K nκ n (2 λ − contains a generalised keyring R ∪ L ∪ · · · ∪ L λ − with 2 λ − K nn + ℓ +1 contains an n -sphere S with atleast 2 λ + 1 n -simplices.Orient S arbitrarily, and orient the L i such that ℓk ( R, L i ) > i . ApplyingLemma 2.5 to the oriented link L = S ∪ L ∪ · · · ∪ L λ − we obtain an n -sphere S withall its vertices on L and meeting each L i in a single n -simplex δ i , which receives oppositeorientations from S and from L i . Set S = S , and for i = 1 , . . . , λ − S i be thecomplex obtained from S i − and L i by omitting the interior of the disc δ i . Then S i is an n -sphere, because it is the result of gluing two discs along their common boundary ∂δ i ,and as a chain we have S i = S + i X j =1 L j (4.1)for i ≥ S i with R , by considering equation (4.1)in the group H n ( R n +1 − R ; Z ). This gives ℓk ( R, S i ) = [ S i ] = [ S ] + i X j =1 [ L j ] = ℓk ( R, S ) + i X j =1 ℓk ( R, L j ) . As in the proof of Lemma 2 of Flapan et al. the sequence (cid:0) ℓk ( R, S i ) (cid:1) λ − i =0 is strictlyincreasing, because the linking numbers ℓk ( R, L i ) are all positive. This sequence musttherefore take 2 λ distinct values, and the result now follows from the fact that there areonly 2 λ − k such that | k | < λ . (cid:3) OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 15 The linking number modulo a prime p . To prove Theorem 1.5 we will use thefollowing lemma on sums of subsequences of finite integer sequences, considered mod p .Given an integer sequence ( ℓ , . . . , ℓ m ) we will say that x ∈ Z is a subsequence sum of( ℓ , . . . , ℓ m ) if there is a subset A ⊆ { , . . . , m } such that X i ∈ A ℓ i = x. We allow the possibility that A is empty, which implies that 0 is always a subsequencesum. Then: Lemma 4.1.
Let p ∈ N be prime, and let ( ℓ , . . . , ℓ p − ) be a sequence of integers suchthat no ℓ i is divisible by p . For any s ∈ Z there is a subsequence sum x of ( ℓ , . . . , ℓ p − ) such that x ≡ s mod p . We note that the sequence length p − p − p realises exactly p − p residue classes as subsequencesums. Proof.
For j = 1 , . . . , p − j be the set of mod p residue classes that may be realisedby a subsequence sum of ( ℓ , . . . , ℓ j ). Then Σ = (cid:8) , ℓ (cid:9) , and our goal is to show thatΣ p − = (cid:8) , , . . . , p − (cid:9) . We will do this by showing that | Σ j +1 | ≥ | Σ j | + 1 wheneverΣ j = (cid:8) , , . . . , p − (cid:9) . Since Σ j ⊆ Σ j +1 it suffices to show that there is an element ofΣ j +1 that is not an element of Σ j .Suppose then that Σ j = (cid:8) , , . . . , p − (cid:9) , and consider multiples of ℓ j +1 mod p . Since ℓ j +1 p we have (cid:8) kℓ j +1 | ≤ k ≤ p − (cid:9) = (cid:8) , , . . . , p − (cid:9) ) Σ j ⊇ (cid:8) (cid:9) , so there is some 1 ≤ k ≤ p − kℓ j +1 / ∈ Σ j . Consider the least such k . Thenthere is a (possibly empty, if k = 1) subset A of { , . . . , j } such that X i ∈ A ℓ i ≡ ( k − ℓ j +1 mod p, and setting B = A ∪ { j + 1 } we have X i ∈ B ℓ i ≡ kℓ j +1 mod p. Hence kℓ j +1 belongs to Σ j +1 but not Σ j , and we are done. (cid:3) Proof of Theorem 1.5.
The technique is similar to that used in the proof of Theorem 1.3.By Theorem 1.2 and Corollary 2.7, N is so large that every embedding of K nN in R n +1 contains a generalised keyring R ∪ L ∪ · · · ∪ L p − with 2 p − S with at least 2 p − n -simplices. Orient the link S ∪ L ∪ · · · ∪ L p − as inthe proof of Theorem 1.3, and let S be the n -sphere that results from applying Lemma 2.5to this link.We now consider the linking numbers ℓk ( R, S ) and ℓk ( R, L i ) modulo p . If ℓk ( R, L i ) ≡ i then we are done, so we may assume that all such linking numbers are nonzero OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 16 mod p . Then by Lemma 4.1 there is a subset A ⊆ { , . . . , p − } such that X i ∈ A [ L i ] ≡ − [ S ] mod p, and a subset B ⊆ { p + 1 , . . . , p − } such that X i ∈ B [ L i ] ≡ − [ L p ] mod p. Set C = B ∪ { p } , to obtain a nonempty subset of { p, . . . , p − } such that X i ∈ C [ L i ] ≡ p. We now consider the chains S = S + X i ∈ A L i , S = S + X i ∈ C L i . In the homology group H n ( R n +1 − R ; Z ) we have[ S ] ≡ [ S ] ≡ p, and moreover [ S ] = [ S ], because the linking numbers [ L i ] are all positive and C isnonempty. It follows that at least one of [ S ] and [ S ] is nonzero, and since both chainsrepresent n -spheres we are done. (cid:3) We note that the argument used above does require p to be prime. For q composite, if ℓk ( R, S ) is coprime to q and all linking numbers ℓk ( R, L i ) are equal to the same nontrivialdivisor d of q , then no sphere formed from S and the L i as above will link R with linkingnumber divisible by q . We will therefore use a different strategy in Section 6 to prove thecorresponding result when q may be composite.5. Technical preliminaries II: Triangulations of an M -simplex We now establish some additional technical preliminaries needed to prove Theorem 1.4.For this theorem we will need to work with links containing identically triangulated discs D n with many n -simplices, and to this end we will construct a triangulation of an M -simplex into many M -simplices.5.1. The triangulations.
For ℓ ∈ N let ∆ Mℓ be the M -simplex∆ = [ ℓ e , ℓ e , . . . , ℓ e M +1 ] ⊆ R M +1 , where e , e , . . . , e M +1 are the standard basis vectors. Then: Lemma 5.1.
The family of planes ( j X k = i x k ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ i ≤ j ≤ M ) (5.1) subdivides ∆ Mℓ into ℓ M M -simplices. The symmetry group of this triangulation is thedihedral group D M +1 of order M + 1) , with the action given by permutations of the basisvectors e i that preserve or reverse the cyclic ordering e , e , . . . , e M +1 . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 17 We will call an M -simplex triangulated as in Lemma 5.1 a triangulated M -simplex ofside-length ℓ , and denote it by ∆ M ( ℓ ). Remark . The triangulation ∆ ( ℓ ) is simply the standard division of an equilateraltriangle of side-length ℓ into ℓ equilateral triangles of side-length 1. In this case allsimplices of the triangulation are isometric. However, for M ≥ (2), wherefour of the 3-simplices are regular tetrahedra, and the remaining four are obtained bycutting an octahedron along two of the three planes of symmetry that pass through fourvertices. Remark . The M + 1 cycle (1 2 . . . M + 1) in D M +1 reverses orientation of ∆ Ml if andonly if M is odd, and when M is even the order two elements of D M +1 reverse orientationif and only if M ≡ M ( ℓ ) has an orientation reversing symmetry if and onlyif M Proof.
We proceed by subdividing the simplexΣ Mℓ = { x ∈ R M | ≤ x ≤ x ≤ · · · ≤ x M ≤ ℓ } into ℓ M simplices, and then pull this subdivision back to ∆ Ml . The chief reason for workingwith ∆ Mℓ rather than Σ Mℓ is that the symmetries of the triangulation are more readilyseen.We first observe that for each permutation σ ∈ S M , the set δ σ = { x ∈ R M | ≤ x σ (1) ≤ x σ (2) ≤ · · · ≤ x σ ( M ) ≤ } is an M -simplex, and that the collection of such simplices gives a subdivision of I M into M ! simplices. These simplices are defined by the family of planes { x i = 0 } ∪ { x i = 1 } ∪ { x j − x i = 0 } , and translating these according to Z M ≤ R M we see that the family { x i ∈ Z | ≤ i ≤ M } ∪ { x j − x i ∈ Z | ≤ i < j ≤ M } (5.2)gives a subdivision of all of R M into isometric simplices. The planes bounding Σ Mℓ belongto this family, and it follows that the subdivision of R M restricts to a subdivision of Σ Mℓ .This subdivision must have ℓ M simplices, on purely volumetric grounds.We now pull this triangulation back to ∆ Mℓ via the linear map that sends the vertex e i of ∆ Mℓ to the vertex e i + · · · + e M of Σ Mℓ for i ≤ M , and the vertex e M +1 to thevertex . Let { φ i } be the dual basis to { e i } . Then φ i pulls back to φ + · · · + φ i , andwe see that the family (5.2) pulls back to the family (5.1). This linear map induces anaffine homeomorphism between Σ Mℓ and ∆ Mℓ , and so these planes give us the desiredtriangulation.To see that the symmetry group is D M +1 , we observe that on the plane P x i = ℓ containing ∆ Mℓ , the conditions j X k = i x k ∈ Z and i − X k =1 x k + M +1 X k = j +1 x k ∈ Z OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 18 are equivalent. Thus, each family of planes defining the subdivision may be viewed asa division of a necklace of M + 1 beads into two connected components, and conversely.Symmetries of the triangulation therefore correspond to precisely those permutations ofthe beads that preserve adjacency, giving us D M +1 . (cid:3) Construction 5.4.
For M ≥ n + 1 we define K nM ( ℓ ) to be the subcomplex of ∆ M − ( ℓ )consisting of precisely those simplices lying entirely within the n -skeleton (∆ M − ℓ ) ( n ) ∼ = K nM . Each n -simplex of (∆ M − ℓ ) ( n ) lies in an n -dimensional co-ordinate plane, and isisometric to ∆ nℓ ; intersecting the family of planes (5.1) with this subspace subdivides thissimplex into a ∆ n ( ℓ ). Thus K nM ( ℓ ) is a space homeomorphic to K nM , with each n -simplexof K nM mapping onto a copy of ∆ n ( ℓ ). As such we will call it a triangulated complete n -complex on M vertices of side-length ℓ .5.2. Counting the vertices.
The number of vertices in a ∆ k ( ℓ ) is equal to the numberof non-negative integer solutions to the equation x + x + · · · + x k +1 = ℓ, and the number of vertices in the interior of a ∆ k ( ℓ ) is the number of positive integersolutions to this equation. These numbers are (cid:0) k + ℓk (cid:1) and (cid:0) ℓ − k (cid:1) = (cid:0) ℓ − ℓ − k − (cid:1) respectively.Counting the vertices of a ∆ M ( ℓ ) according to the open simplex of ∆ Mℓ that they belongto we find that it has M X k =0 (cid:18) M + 1 k + 1 (cid:19)(cid:18) ℓ − ℓ − k − (cid:19) = (cid:18) ℓ + MM (cid:19) (5.3)vertices (the two sides are the co-efficient of x ℓ in (1 + x ) M +1 (1 + x ) ℓ − = (1 + x ) ℓ + M ).Of particular interest is the number of vertices belonging to K n n +4 ( ℓ ), as this complexis homeomorphic to K n n +4 , and may be used to construct links in which each componenthas many n -simplices. Setting M = 2 n + 3 in equation (5.3), and truncating the sum at k = n , we therefore find that K n n +4 ( ℓ ) has a total of V ( n, ℓ ) = n X k =0 (cid:18) n + 4 k + 1 (cid:19)(cid:18) ℓ − k (cid:19) (5.4)vertices.For a more tractable bound, observe that the triangulated simplex ∆ n ( ℓ ) has ℓ n n -simplices, each with n + 1 vertices, and so has at most ( n + 1) ℓ n vertices. The complex K n n +4 ( ℓ ) contains (cid:0) n +4 n +1 (cid:1) such triangulated simplices, and therefore V ( n, ℓ ) ≤ ( n + 1) (cid:18) n + 4 n + 1 (cid:19) ℓ n (this also follows from the inequalities (cid:0) n +4 k +1 (cid:1) ≤ (cid:0) n +4 n +1 (cid:1) and (cid:0) ℓ − k (cid:1) ≤ ℓ n for k ≤ n ). Stirling’sformula m ! ∼ √ πm ( m/e ) m leads to the asymptotic formula (cid:0) mm (cid:1) ∼ m / √ πm , and hence( n + 1) (cid:18) n + 4 n + 1 (cid:19) = ( n + 1)( n + 2) n + 3 (cid:18) n + 2) n + 2 (cid:19) ∼ r nπ n +2 = C √ n n . Consequently, asymptotically V ( n, ℓ ) grows no faster than C √ n (4 ℓ ) n . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 19 PSfrag replacements x x L Figure 4.
Illustrating the construction of the disc D of Lemma 6.1 inthe case n = 2, ℓ = 4. A line L with irrational slope α > never passesthrough the intersection of two such lines. We take D to be the union ofthe 2-simplices intersecting L (shaded grey). The disc D contains at least ℓ n -simplices (here at least 4), since it must include at least one from eachhorizontal slice. 6. Linking number mod q The goal of this section is to prove Theorem 1.4, which we recall states that given q ∈ N ,embeddings of sufficiently large complete n -complexes in R n +1 contain 2-component linkswith linking number a nonzero multiple of q . Before proving this theorem we need onemore technical lemma: Lemma 6.1.
Let R be a positive integer. For ℓ sufficiently large ∆ n ( ℓ ) contains a trian-gulated disc D with r ≥ R n -simplices ∆ , . . . , ∆ r , which may be labelled such that D ij = j [ k = i ∆ k is a disc for any ≤ i ≤ j ≤ r . The conclusion holds for ℓ ≥ R , so the side lengthrequired grows at most linearly with R .Proof. Write Σ n ( ℓ ) for the n -simplex Σ nℓ subdivided by the family of planes given by equa-tion (5.2). Then Σ n ( ℓ ) and ∆ n ( ℓ ) are simplicially isomorphic, so it suffices to constructa suitable disc D in Σ n ( ℓ ). We will construct D as the union of the n -simplices of Σ n ( ℓ )that meet a suitably chosen line L in R n . The case n = 2, ℓ = 4 is illustrated in Figure 4.Since R is infinite dimensional as a vector space over Q , we may choose 0 < α < · · · <α n = 1 such that { α , . . . , α n } is linearly independent over Q . Write α = ( α , . . . , α n ),and let L be the line L = { t α : t ∈ R } . Each plane in the family (5.2) may be written in OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 20 the form c T x = u , where c ∈ Z n and u ∈ Z , and the linear independence of { α , . . . , α n } over Q may be used to show that(1) L meets each plane in the family (5.2) transversely; and(2) each point of L other than lies on at most one plane in this family.Together these facts imply that, with the exception of simplices containing , L can meetonly n - and ( n − n ( ℓ ), and that if it intersects an n -simplex at all it mustintersect it in its interior.Observe that the line segment { t α : 0 ≤ t ≤ ℓ } is contained in Σ nℓ , and cuts each plane x n = k for k = 1 , . . . , ℓ . Consequently L must pass through at least one n -simplex ofΣ n ( ℓ ) lying in the slice { x : k − ≤ x n ≤ k } for each 1 ≤ k ≤ ℓ , and so passes through atleast ℓ n -simplices of Σ n ( ℓ ). Suppose that L passes through exactly r n -simplices of Σ n ( ℓ ),and label them consecutively ∆ , . . . , ∆ r in the order in which they are encountered whentracing L in the direction α . We claim that D ij = j [ k = i ∆ k is a disc for any 1 ≤ i ≤ j ≤ ℓ , from which the result follows.Since the open ray { t α : t > } only meets n - and ( n − n ( ℓ ), consecutive n -simplices ∆ k and ∆ k +1 must intersect in an ( n − d ≥ k and ∆ k + d are separated by at least two planes from the family (5.2), and someet in at most an ( n − D ii = ∆ i is a disc, and D i,k +1 is the result ofgluing D ik and ∆ k +1 along the ( n − k ∩ ∆ k +1 , it follows by induction that D ij is a disc, as claimed. (cid:3) We now prove Theorem 1.4. The argument again proceeds by converting a suitablylarge generalised keyring to a 2-component link, but now we require additionally that thekeys of the keyring are copies of K nn +2 ( q ). Our underlying approach is similar to that ofFleming [5, Theorem 3.1], but differs from his in the size of the keys and the method usedto combine them to form the second component of the link. Proof of Theorem 1.4.
We show that the result holds for N = 4 q V ( n, q ) + n + (cid:24) q − n (cid:25) + 1 , where V ( n, q ) is given by (5.4) and equals the number of vertices belonging to K n n +4 ( q ).Since V ( n, q ) ≤ ( n + 1) (cid:0) n +4 n +1 (cid:1) q n , we conclude that N grows no faster than C ( n +1) (cid:0) n +4 n +1 (cid:1) q n +2 .Given an embedding of K nN in R n +1 , let C , . . . , C q be disjoint copies of K n n +4 ( q )contained in K nN , and use the remaining n + ⌈ (4 q − /n ⌉ + 1 vertices and Corollary 2.7 toconstruct an n -sphere L with at least 4 q n -simplices. The complex C i is homeomorphicto K n n +4 , and so by Taniyama [15] contains a two component link J i ∪ X i such that ℓk ( J i , X i ) = 0, and each component is a copy of K nn +2 ( q ). Applying Lemma 3.2 to thelink L ∪ J ∪ · · · ∪ J q ∪ X ∪ · · · ∪ X q we obtain a generalised keyring R ∪ L ∪ · · · ∪ L q ,where ℓk ( R, L i ) = 0 for each i , and each L i is a copy of K nn +2 ( q ). We will use R as one OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 21 component of our link, and we will seek to construct the second as a connect sum of someof the L i . In what follows we therefore consider homology classes in H n ( R n +1 − R ; Z ).Orient the L i such that ℓk ( R, L i ) = [ L i ] is positive for each i , and for 1 ≤ k ≤ q consider the values of the sums P ki =1 [ L i ] mod q . Since there are q sums and q possiblevalues modulo q , by the Pigeonhole Principle there must either be a sum that is zeromod q , or else two sums that are equal modulo q . In either case we obtain integers a, b satisfying 1 ≤ a ≤ b ≤ q such that b X i = a [ L i ] ≡ q. From now on we restrict our attention to the spheres L a , . . . , L b .Our construction now departs from that of Fleming. Each component L i is a copy of K nn +2 ( q ), and as such has n + 2 faces which are triangulated n -simplices of sidelength q .We claim that it is possible to choose distinct faces δ i , δ ′ i of L i , each a copy of ∆ n ( q ),and orientation reversing simplicial isomorphisms ψ i : δ i → δ ′ i +1 . For n δ i , δ ′ i of L i arbitrarily, since in this case ∆ n ( q )has both orientation preserving and reversing symmetries, by Remark 5.3. However, for n ≡ δ a and using the fact that∆ n ( q ) has at least one face of each orientation to choose δ ′ i +1 based on the choice of δ i .The face δ i +1 of L i may then be chosen arbitrarily from those left.By Lemma 6.1 each face δ i ∼ = ∆ n ( q ) contains a triangulated disc D i with r ≥ q n -simplices ∆ i , . . . , ∆ ir , such that ( D i ) cd = d [ k = c ∆ ik is a disc for each 1 ≤ c ≤ d ≤ r . Let φ i be the restriction of ψ i to D i , let D ′ i +1 = φ i ( D i ),and for 1 ≤ j ≤ r let P ij be the oriented sphere satisfying P ij ∩ L i = ∆ ij , P ij ∩ L i +1 = φ i (∆ ij )that results from applying Corollary 2.2 to the pairs ( L i , D i ) and ( L i +1 , D ′ i +1 ).For 1 ≤ k ≤ r we now consider the sums P kj =1 [ P ij ] modulo q . Since there are q possible values mod q and at least q sums we may again choose integers c i , d i satisfying1 ≤ c i ≤ d i ≤ r such that d i X j = c i [ P ij ] ≡ q. Let Q i = P d i j = c i P ij . Then Q i represents an n -sphere with all its vertices on L i ∪ L i +1 andsatisfying Q i ∩ L i = ( D i ) c i d i , Q i ∩ L i +1 = φ i (( D i ) c i d i ) , ℓk ( R, Q i ) ≡ q. If ℓk ( R, Q i ) = 0 for some i then we are done by setting S = Q i , so we may assumethat in fact ℓk ( R, Q i ) = 0 for all i . In that case we let S be the complex obtained from L a , . . . , L b and Q a , . . . , Q b − by omitting the interiors of the discs Q a ∩ L a , . . . , Q b − ∩ L b − OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 22 and Q a ∩ L a +1 , . . . , Q b − ∩ L b . Then S is a connect sum of n -spheres, hence an n -sphere,and as a chain we have S = b X i = a L i + b − X i = a Q i . It follows that [ S ] = b X i = a [ L i ] + b − X i = a [ Q i ] = b X i = a [ L i ] > , and since also P bi = a [ L i ] ≡ q we are done. (cid:3) Remark . For n = 1 the auxiliary sphere S of Lemma 2.5 is not needed to constructthe keyring, reducing the number of vertices required in this case to4 q V (1 , q ) = 4 q (6 + 15( q − q (5 q − , as given after the statement of the theorem.7. An alternate proof of Theorem 1.3
To further illustrate the applications of the triangulations of Section 5 we give a secondproof of Theorem 1.3, without the polynomial bound on the number of vertices required.Namely, we show that given ℓ ∈ N , for N sufficiently large every embedding of K nN in R n +1 contains a 2-component link with linking number at least ℓ in absolute value.The proof we give is modelled on Flapan’s original proof [2] of the corresponding resultfor n = 1. Her argument is based on combining 2-component links with “sufficientlymany vertices”, and for n ≥ n -simplices of sufficient sidelength. The side length available will typically shrink when two components are combined(unlike the number of vertices, which typically goes up), and consequently this changeleads to a significant change in the growth of the number of vertices required.7.1. Splicing links.
In this section we establish higher dimensional analogues of Lem-mas 2 and 1 of Flapan [2]. These are Lemmas 7.2 and 7.3 below respectively. In prepa-ration for this we need an additional technical lemma on triangulated n -simplices. Lemma 7.1.
Deleting an arbitrary M -simplex from a triangulated M -simplex of side-length ℓ leaves a triangulated M -simplex of side-length at least ⌊ M ℓ/ ( M + 1) ⌋ .Proof. Let δ be the deleted simplex, and let x be a point in the interior of δ . In barycentricco-ordinates on ∆ Mℓ we have x = ℓ M +1 X i =1 t i e i , and since P t i = 1 we must have t i ≤ / ( M + 1) for some i . Let ∆ be the intersectionof ∆ M ( ℓ ) with the halfspace x i ≥ ⌈ ℓ/ ( M + 1) ⌉ . Then ∆ is a triangulated M -simplexcontained in ∆ Mℓ , and ∆ does not contain δ because ∆ does not contain x . Moreover, ∆has side-length ℓ − (cid:24) ℓM + 1 (cid:25) = (cid:22) ℓ − ℓM + 1 (cid:23) = (cid:22) M ℓM + 1 (cid:23) , OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 23 so we are done. (cid:3) Lemma 7.2.
Let X ∪ Y ∪ X ∪ Y be a -component link contained in some embedding of K nN in R n +1 . Suppose that for some orientation of X ∪ Y ∪ X ∪ Y we have ℓk ( X , Y ) ≥ and ℓk ( X , Y ) = p ≥ , and suppose also that each component contains a triangulated n -simplex of side-length ℓ with ℓ n ≥ p . Then K nN contains disjoint n -spheres L , Z and W such that (1) ℓk ( L, Z ) = p ≥ and ℓk ( L, W ) = p ≥ p for some orientation of the link L ∪ Z ∪ W ; (2) L contains a triangulated n -simplex of side-length at least ⌊ nℓ/ ( n + 1) ⌋ ; (3) Z is equal to either X or Y ; (4) W is equal to either X or Y .Proof. As in Flapan [2], if ℓk ( X , Y ) is non-zero we may set L = X , Z = Y , and W = Y ;and if ℓk ( Y , X ) is non-zero we may set L = Y , Z = X , and W = X . So in what followswe may assume that ℓk ( X , Y ) = ℓk ( X , Y ) = 0.Let D i be a ∆ n ( ℓ ) contained in X i , for each i , and let φ : D → D be a simplicialisomorphism. After reversing orientation on both X and Y if necessary we may assumethat φ reverses orientation, and so we may apply Corollary 2.2 to the pairs ( X , D ) and( X , D ). We label the resulting spheres P , . . . , P ℓ n as in the statement of the corollary,and following Flapan the equation[ X ] + [ X ] + ℓ n X j =0 [ P j ] = 0holds in the n th homology group H n ( R n +1 − Y ; Z ).By our assumption that ℓk ( X , Y ) = 0 we have [ X ] = 0 in H n ( R n +1 − Y ; Z ), so0 < p = [ X ] = − ℓ n X j =0 [ P j ] . The right hand side consists of ℓ n +1 > p terms, so for some index q we must have [ P q ] ≥ P q ] = 0 in H n ( R n +1 − Y ; Z ).If [ P q ] is non-zero in H n ( R n +1 − Y ; Z ) then we construct L from P q and X by deletingthe interior of the disc X ∩ P q . L is the connect sum of the n -spheres P q and X , and sois itself an n -sphere. As a chain we have L = P q + X , and therefore[ L ] = [ P q ] + [ X ] ≥ p in H n ( R n +1 − Y ; Z ) , [ L ] = [ P q ] + [ X ] = [ P q ] = 0 in H n ( R n +1 − Y ; Z ) . So we obtain the desired link by letting Z = Y and W = Y , and re-orienting Z ifnecessary so that ℓk ( L, Z ) is positive.If [ P q ] = 0 in H n ( R n +1 − Y ; Z ) then we construct L from X , X and P q by deletingthe interiors of the discs X i ∩ P q . Clearly, L is again an n -sphere. As a chain we have L = X + P q + X , and therefore[ L ] = [ X ] + [ P q ] + [ X ] = [ P q ] + [ X ] ≥ p in H n ( R n +1 − Y ; Z ) , [ L ] = [ X ] + [ P q ] + [ X ] = [ X ] ≥ H n ( R n +1 − Y ; Z ) . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 24 So we obtain the desired link by letting Z = Y and W = Y .In every case above Z was equal to either X or Y , and W was equal to either X or Y . To complete the proof we must show that L contains a triangulated n -simplexof side-length at least ⌊ nℓ/ ( n + 1) ⌋ . If q = 0 then L contains D and we are done, andotherwise L contains D \ ( X ∩ P q ) and we are done by Lemma 7.1. (cid:3) Lemma 7.3.
Let L ∪ Z ∪ W be a -component link contained in some embedding of K nN in R n +1 , and suppose that for some orientation of L ∪ Z ∪ W we have ℓk ( L, Z ) = p > , ℓk ( L, W ) = p > . Suppose that Z and W contain triangulated simplices ∆ Z and ∆ W of side-length ℓ , with ℓ n ≥ p + p , and that there is an orientation reversing simplicialisomorphism φ : ∆ Z → ∆ W . Then K nN contains an n -sphere J disjoint from L such that (1) ℓk ( L, J ) ≥ p + p for some orientation of L ∪ J ; (2) J contains a triangulated n -simplex of side-length at least ⌊ nℓ/ ( n + 1) ⌋ .Proof. As in the proof of Lemma 7.2 we apply Corollary 2.2 to the pairs ( Z, ∆ Z ) and( W, ∆ W ), obtaining spheres P , . . . , P ℓ n . In the homology group H n ( R n +1 − L ; Z ) wehave the equation [ Z ] + [ W ] + ℓ n X j =0 [ P j ] = 0 , so that p + p = [ Z ] + [ W ] = − ℓ n X j =0 [ P j ] . As in the proof of Lemma 7.2 above, the right-hand side has ℓ n + 1 > p + p terms, sothere must be an index q such that [ P q ] ≥
0. Let J be the n -sphere obtained from Z , P q and W by deleting the interiors of the discs P q ∩ Z and P q ∩ W . Then J is disjoint from L by Remark 2.3, and as a chain J = Z + P q + W , so[ J ] = [ Z ] + [ P q ] + [ W ] ≥ p + p in H n ( R n +1 − L ; Z ). Condition (2) above holds by the same argument as in Lemma 7.2,and the result follows. (cid:3) Combining Lemmas 7.2 and 7.3 we obtain the following:
Corollary 7.4.
Let X ∪ Y ∪ X ∪ Y be a -component link contained in some embeddingof K nN in R n +1 . Suppose that (1) for some orientation of X ∪ Y ∪ X ∪ Y we have ℓk ( X , Y ) ≥ and ℓk ( X , Y ) = p ≥ ; (2) each component contains a triangulated n -simplex of side-length ℓ with ℓ n ≥ p ; (3) either n , or X and Y each contain two such triangulated n -simplices,one of each possible orientation.Then K nN contains disjoint n -spheres L and J , each containing a triangulated n -simplexof side length at least ⌊ nℓ/ ( n + 1) ⌋ , and such that ℓk ( L, J ) ≥ p + 1 .Proof. The hypotheses of Lemma 7.2 are satisfied, so we obtain a three component link L ∪ Z ∪ W satisfying the conditions given in that Lemma. These conditions imply the OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 25 hypotheses of Lemma 7.3, except perhaps the condition that ℓ n ≥ p + p and the conditionthat φ may be chosen to reverse orientation.If the hypothesis ℓ n ≥ p + p does not hold then we must have p + p > p , whichimplies p i ≥ p + 1 for some i . So if this occurs we are done by simply letting J be either Z or W , as appropriate.To see that the condition on φ is satisfied we use our third hypothesis above. If n n ( ℓ ) has an orientation reversing symmetry, and otherwise Z is equal toeither X or Y , and so contains a ∆ n ( ℓ ) of each orientation. We may therefore choose ∆ Z and ∆ W to have opposite orientations, and apply Lemma 7.3 to get the desired result. (cid:3) Theorem 1.3, revisited.
Using the results of the previous section we re-proveTheorem 1.3 in the following weakened form.
Theorem 7.5.
Given λ ≥ , let µ = l n p λ − m , and suppose that N is sufficientlylarge that K nN contains disjoint copies of K n n +4 (2 i µ ) for i = 0 , . . . , λ − , and an additionaldisjoint copy of K n n +4 (2 λ − µ ) . Then every embedding of K nN in R n +1 contains a two-component link L ∪ J such that, for some orientation of the components, ℓk ( L, J ) ≥ λ .Proof. Given an embedding of K nN in R n +1 , let C , . . . , C λ be disjoint subcomplexes of K nN such that C is a K n n +4 (2 λ − µ ), and C i is a K n n +4 (2 λ − i µ ) for i = 2 , . . . , λ . Each C i ishomeomorphic to K n n +4 , and so by Taniyama [15] contains a two component link S i ∪ T i which we may orient such that ℓk ( S i , T i ) ≥
1. We will use these to inductively constructlinks L i ∪ J i such that(1) ℓk ( L i , J i ) ≥ i ;(2) all vertices of L i ∪ J i lie in C ∪ · · · ∪ C i (and so L i ∪ J i is disjoint from C j for j > i );(3) for i < λ the spheres L i and J i each contain a triangulated n -complex of side-lengthat least 2 λ − i − µ .The link L λ ∪ J λ is then the required link.Each component S i , T i is isomorphic to the boundary of a triangulated ( n + 1)-simplexof side-length equal to that of C i , and as such has n +1 faces which are each a triangulated n -simplex of this same side-length. For the base case we may therefore simply let L ∪ J = S ∪ T .Given 1 ≤ i ≤ λ −
1, suppose that we have constructed L i ∪ J i but not yet L i +1 ∪ J i +1 .Let ℓk ( S i , T i ) = p ≥ i . If p ≥ λ then we simply set L j ∪ J j = S i ∪ T i for j ≥ i and the construction is complete, so suppose that p < λ . Then every component ofthe link S i +1 ∪ T i +1 ∪ L i ∪ J i contains a triangulated n -simplex of side-length at least ℓ = 2 λ − i − µ ≥ µ , and ℓ satisfies ℓ n ≥ µ n ≥ λ − ≥ p . Moreover, as the boundaryof a K nn +1 ( ℓ ), each component of S i +1 ∪ T i +1 must contain at least one ∆ n ( ℓ ) face of eachorientation. Working entirely within the K nM spanned by the vertices of C ∪ · · · ∪ C i +1 we may therefore apply Corollary 7.4 to obtain a 2-component link L i +1 ∪ J i +1 satisfying ℓk ( L i +1 , J i +1 ) ≥ p + 1 ≥ i + 1.Each component of L i +1 ∪ J i +1 contains a triangulated n -simplex of side-length at least (cid:22) nℓn + 1 (cid:23) = (cid:22) λ − i − nµn + 1 (cid:23) . OME RAMSEY-TYPE RESULTS ON INTRINSIC LINKING OF n -COMPLEXES 26 Now nn +1 ≥ , so for i < λ − λ − i − µ is an integer satisfying2 λ − i − nµn + 1 ≥ λ − i − µ λ − i − µ, and therefore (cid:22) nℓn + 1 (cid:23) = (cid:22) λ − i − nµn + 1 (cid:23) ≥ λ − i − µ = 2 λ − ( i +1) − µ. This establishes condition (3) above when i + 1 < λ , completing the inductive step. (cid:3) References [1] J. H. Conway and C. McA. Gordon. Knots and links in spatial graphs.
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