Knots and Links in Spatial Graphs: a Survey

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Discrete Mathematics 302 (2005) 225–242 www.elsevier.com/locate/disc

Knots and links in spatial graphs:A survey J.L. Ramírez Alfonsín Université Pierre et Marie Curie, Paris 6, Equipe Combinatoire - Case 189, 4 Place Jussieu Paris 75252 Cedex 05, France

Received 7 December 2001; received in revised form 15 May 2003; accepted 22 July 2004 Available online 8 August 2005

Abstract A spatial representation, R(G), of a graph G, is an embedded image of G in R3. A set of cycles in R(G) can be thought of as a set of simple closed curves in R3 and thus they may be regarded as a link in R3. A recent area of research investigates the dependence (or independence) of the link types on the structure of the abstract graph G itself rather than on speciﬁc spatial representations of G.In this article, we survey what is known today. © 2005 Elsevier B.V. All rights reserved.

Keywords: Knot; Link; Spatial graphs

1. Introduction

By a graph we mean a ﬁnite undirected graph (loops and multiple edges allowed). A graph is complete if every pair of vertices is adjacent. A complete graph with n vertices is denoted by Kn. A graph is said to be bipartite if its set of vertices can be partitioned into two disjoint sets A and B (bipartition) such that no two vertices in the same set are adjacent. A complete bipartite graph is a bipartite graph with bipartition A and B in which each vertex of A is joined to each vertex in B. A complete bipartite graph with |A|=m and |B|=n is denoted by Km,n.Acycle in a graph G is a connected subgraph G of G such that every vertex of G has exactly two incident edges. In particular, a cycle that contains every vertex of G is called a Hamiltonian cycle of G; see [8] for further graph theory details. A spatial representation R(G), of a graph G, is the embedded image of G in R3, that is, the vertices of G are distinct points in 3-dimensional space and the edges are simple

E-mail address: [email protected].

Fig. 1. A R(K6) with two concatenated cycles.

Jordan curves between them in such a way that any two curves are either disjoint or meet at a common end point. Throughout this paper we consider tame spatial representations, i.e., spatial representations that have piecewise linear edges. The cycles in R(G) can be thought of as simple closed curves in R3. Hence, we may regard a set of spatial cycles as a link in R3. The study of the dependence (and independence) of the link (and knot) types contained in spatial representations has been the attention of recent research. These investigations have generated a number of research papers and, at present, there are not comprehensive or accessible sources summarizing the progress on it. This manuscript has a goal to survey what is known today. In Section 2 we describe and summarize the results in relation with concatenated graphs. Section 3 discusses the results on self-knotted graphs. Section 4 is dedicated to realizable embeddings. In Section 5 we survey what is known on linear embeddings. Finally, we give an Appendix with some basic deﬁnitions on knot theory needed throughout this paper.

2. Concatenated graphs

In 1973, Bothe [9] raised the following question:1 Consider a set of six distinct points in the Euclidean 3-dimensional space R3 and assume that each pair of these points is connected by a simple curve such that no two of these curves have a point in common—except for their endpoints, of course. Is it true that in this spatial figure we can always find a pair of simple closed curves that are ‘concatenated’? Two curves are called concatenated if they cannot be embedded in disjoint (topological) balls. Fig. 1 shows such spatial figure with exactly one pair of concatenated triangles. Sachs [59] answered the above question positively (this was also answered by Conway and Gordon [16]).

1 At the Eger Conference, W. Moser informed H. Sachs that the same problem had also been raised by his late brother Leo Moser about 15 years earlier (unpublished). J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 227

Theorem 2.1 (Conway and Gordon [16], Sachs [59]). Any R(K6) contains a set of two 2 cycles which is isotopic to the link 21.

Sachs proved Theorem 2.1 by using the unlinking number of links while Conway and Gordon proved it by showing that every R(K6) contains an odd number of pairs, among the set of all pairs of disjoint cycles in K6, with odd linking number (using the mod 2 linking number of links); see also [5,6] for related discussions. The degree of concatenation of two disjoint cycles in a R(G) is the minimal number of permeations necessary to disconcatenate them (the permeation of the arcs of the same cycle is counted with multiplicity 2). The degree of concatenation of R(G) is the sum of the degrees of concatenation over all unordered pairs of disjoint cycles. Finally, the degree of concatenation of G is the minimal degree of concatenation of a R(G). G is said to be discatenable if its degree of concatenation is zero. Sachs posed the following question [59, Problem 1]:Let D be the class of all discatenable graphs, can the class D be characterized by a ﬁnite set of forbidden subgraphs? Theorem 2.1 implies that K6 ∈/ D. Nešetˇril and Thomas [42] answered the above question positively even in a stronger sense (recall that H is a minor of a graph G if there exists a subgraph G of G which can be contracted onto H).

Theorem 2.2 (Nešetˇril and Thomas [42]). Let k be a positive integer and let Dk be the class of all graphs with degree of concatenation less or equals to k. Then, there exists a Gk,...,Gk G ∈ D Gk set of graphs 1 n(k) with the following property: k if and only if no i is a minor of G.

Nešetˇril and Thomas proved Theorem 2.2 by using the following outstanding result (known as Wagner’s conjecture) due to Robertson and Seymour [51].

Theorem 2.3 (Robertson and Seymour [51]). Every class of ﬁnite graphs which is closed under minors can be determined by a ﬁnite set of forbidden minors.

Presently, no bound can be deduced from the Robertson–Seymour argument (which is based on the theory of well-quasiorderings). In a series of three papers by Robertson et al. [54–56] (see also [52]) strengthened Gk,...,Gk k = Theorem 2.2 by characterizing the set of graphs 1 n(k) when 1. Before stating 2 their result, recall that the Petersen family consists of all graphs obtainable from K6 by the so-called Y − and − Y operations (Y − means deleting a vertex of degree 3, and making its neighboring vertices mutually adjacent. − Y is the reverse operation applied to a triangle). Fig. 2 shows the Petersen family.

Theorem 2.4 (Robertson et al. [53,56]). A graph G is discatenable if and only if it contains as a minor none of the seven Petersen family graphs.

2 The name of this family comes from the fact that it contains the well-known Petersen graph. 228 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

Petersen Graph

Fig. 2. The Petersen family.

Motivated by estimating the maximum multiplicity of the second eigenvalue of Schrödinger operators, Colin de Verdière [14,15] introduced an interesting new invariant, (G), for a graph G, based on spectral properties of matrices associated with a graph G. Colin de Verdière showed that the invariant is monotone under taking minors (i.e., if H is a minor of G then (H )(G)), that (G)1 if and only if G is a disjoint union of paths, that (G)2 if and only if G is outerplanar, and that (G)3 if and only if G is planar. Since any of the graphs in the Petersen family has (G)5 (which was proved by Bacher and Colin de Verdière in [2]), it follows that if (G)4 then G is discatenable. Lovász and Schrijver [29] have proved the reverse implication (conjectured by Robertson and Seymour).

Theorem 2.5 (Lovász and Schrijver [29]). G is discatenable if and only if (G)4.

A key ingredient in the proof of Theorem 2.5 is a Borsuk-type theorem on the existence of antipodal links. Theorem 2.5 gives a spectral characterization (beside the combinatorial characterization, in terms of minors) of the topologically deﬁned class of discatenable graphs. A triple link is a nonsplit link of three components. We say that a graph G is intrinsically triple linked if every spatial representation of G contains a triple link. Chan et al. [23] J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 229 proved that K3,3,3 is not intrinsically triple linked and conjectured that K9 is intrinsically triple linked. Flapan et al. [21] disproved this conjecture.

Theorem 2.6 (Flapan et al. [21]). K10 is intrinsically triple linked and K9 is not intrinsically triple linked.

A graph is intrinsically n-linked if every spatial representation of G contains a nonsplit link of n components. Flapan et al. [20] studied the intrinsically n-linked graphs.

Theorem 2.7 (Flapan et al. [20]). For any natural number n, there always exist an intrinsically (n + 1)-linked graph.

The graphs in Theorem 2.7 are exhibited explicitly. Moreover, they showed that each of these graphs is minor minimal in the sense that every minor of it has an embedding in R3 that contains no nonsplit n-component link. In [7] Bokor and Johnson extended the concept of intrinsically n-linked as follows. A graph G is (s, t)-intrinsically linked if every spatial representation of G contains a nontrivial link whose components are an s-cycle and a t-cycle (a r-cycle is a cycle of length r). Such a link is called an (s, t)-link.

Theorem 2.8 (Bokor and Johnson [7]). Kn,n is (4, 2n − 4) intrinsically linked and Kn,n,1 is (4, 2n − 3)-intrinsically linked.

Johnson and Johnson [25] generalized Theorem 2.8. A graph G with n vertices that is (s, t)-intrinsically linked for all s + t = n for which G has s- and t-cycles is a completely linked graph.

Theorem 2.9 (Johnson and Johnson [25]). For any n4, Kn,n is completely linked.

As a consequence of Theorem 2.9, Johnson and Johnson [25] obtained the following results:

Corollary 2.10. For n3, and any even integer 4k 2n − 2, Kn,n,1 is (k, 2n − k + 1)- intrinsically linked.

Corollary 2.11. For n3, K2n+1 is completely linked.

Lemma 2.12. For n3, Kn,n,1,1 is (k − 1, 2n − k + 3)-intrinsically linked for all even k such that 4k 2n.

Corollary 2.13. For n3, K2n is completely linked.

Notice that Corollary 2.13 contains Theorem 2.1 when n = 3. Corollaries 2.11 and 2.13 have also been proved by Dochterman et al. [18]. Recently, Flapan [19] showed that for every m ∈ N there exists an n ∈ N such that every spatial representation of the complete graph on n vertices contains a link of two components whose linking number is at least m. 230 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

3 Fig. 3. An special embedding of K7 in R .

We close this section by mentioning that B˝ohme [4] studied the following question (posed by Sachs in [58]):how many edges a discatenable graph on n vertices can have?

3. Self-knotted graphs

A graph G is called self-knotted if every R(G) contains a knotted cycle. In the same pio- neer paper in which Theorem 2.1 is proved, Gordon and Conway also showed the following result.

Theorem 3.1 (Conway and Gordon [16]). Every R(K7) is self-knotted. Proof (Sketch). Given a R(K7) deﬁne ∈ Z2 by = (C) where (K) denote the Arf-invariant of knot K and the summation being over all 360 = 1 6! Hamiltonian cycles 2 C in K7. It is shown that is invariant under crossing changes. Since = (mod 2) is unaffected by crossing changes, then must be the same for every R(K7). In particular, if is equals to 1 for any speciﬁc R(K7) then is equals to 1 for every R(K7). It is routine to verify that the R(K7) shown in Fig. 3 has all Hamiltonian cycles unknotted (and thus with Arf-invariant equal to zero) except one isotopic to the trefoil (and thus with Arf-invariant equal to one). Finally, since = 1 for every R(K7) then for a given R(K7), it cannot be the case that all the Hamiltonian cycles in the embedding are unknotted (otherwise, we may have a R(K7) with = 0). Therefore, every R(K7) is self-knotted.

Note that Theorem 3.1 says nothing about the knot type. There may exist a R(K7) with all Hamiltonian cycles unknotted (with Arf-invariant equal to zero) except one isotopic to, say, the ﬁgure-eight (with Arf-invariant equal to one). In 1988, Shimabara [60] generalized Conway–Gordon’s method to prove the following two results:

Theorem 3.2 (Shimabara [60]). The complete bipartite graph K5,5 is self-knotted.

Theorem 3.3 (Shimabara [60]). Every R(K5,5 −{e}), R(K4,4,1) and R(Km,m) (m5) has an even number of Hamiltonian cycles whose Arf-invariant are one. J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 231

a a

b b

c 1 5 1 5 4 2 3 2 3 4

d c

e d

3 Fig. 4. Embeddings of K5,5 and K4,5 in R .

Fig. 4 illustrates both a R(K5,5) containing a knotted cycle isotopic to the trefoil (formed by sequence [5,a,3,d,1,b,4,e,2,c,5]) and a Hopf link (formed by sequences [a,5,c,2, d,3,a] and [b, 1,e,4,b]) and a R(K4,5) having no knotted cycles but containing a Hopf link (formed by the sequences [a,3,c,1,a] and [b, 4,d,2,b]). F Fc F Let SK be the family of all self-knotted graphs. Let SK be the complement of SK F Fc in the family of all ﬁnite graphs. Motwani et al. [34] studied the sets SK and SK.

Proposition 3.4 (Motwani et al. [34, Lemma 2]). Suppose that G is a graph obtained fromGbya − Y transformation.

(i) If G ∈ FSK then G ∈ FSK and G ∈ Fc G ∈ Fc (ii) If SK then SK.

Proposition 3.4 and Theorem 3.1 imply that the application of the −Y transformation to K7 would leads to another self-knotted graph. By repeatedly applying this transformation until there are no triangles left, one may obtain 13 more graphs which are self-knotted. Kohara and Suzuki [28] analyzed this set of graphs.

Theorem 3.5 (Kohara and Suzuki [28]). The 13 graphs that are obtained from K7 by − Y Fc repeatedly applying transformation are forbidden for SK.

Theorem 3.5 was also pointed out by Motwani et al. [34]. A graph G is said to be critical with respect to self-knotted if and only if G ∈ FSK, the Fc minimum degree of G is at least three and any proper subgraph H of G belongs to SK.In [28], Kohara and Suzuki also discussed the set of all critical graphs. denoted by C(FSK). O(Fc ) ⊂ C(F ) O(Fc ) They observed that SK SK where SK denotes the obstruction set for Fc SK. It follows immediately from Theorems 2.1 and 3.1 that 232 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

R( ) ( n ) n (1) Every Kn contains at least 6 pairs of disjoint cycles that are concatenable ( 6) and R( ) ( n ) n (2) Every Kn contains at least 7 7-cycles that are nontrivial ( 7). It is an interesting problem to decide whether or not there exists a R(Kn) containing exactly ( n ) ( n ) 6 pairs of disjoint cycles that are concatenable (resp. containing exactly 7 7-cycles that are nontrivial). Note that this is true for n = 6 (resp. n = 7); see Figs. 1 and 3. Otsuki [45] proved, by introducing the concept of canonical book representation, that the lower bounds are best possible.

Theorem 3.6 (Otsuki [45]). There exists a R(Kn) such that the number of pairs of disjoint ( n ) 2 n R( ) linked 3-cycles is exactly 6 . All of the links are 21 for 6. There exists a Kn such ( n ) that the number of knotted 7-cycles is exactly 7 . All of the knotted 7-cycles are trefoils for n7.

The embeddings stated in Theorem 3.6 are given explicitly. Thomas [65, p. 212] asked whether K1,1,3,3 is self-knotted. A positive answer was given by Foisy [22] by adapting Conway and Gordon’s method. In view of Theorem 2.5, it is tempting to ask whether there is a relationship between a not self-knotted graph G and (G)5. We ﬁnally mention the following nice generalization of Theorem 3.1 in higher dimensions, due to Taniyama [63]:any embedding of the n-skeleton of a (2n+3)-dimensional sim- plex into the (2n+1)-dimensional sphere contains a nonsplitable link of two n-dimensional sphere.

4. Realizable embeddings

Let n be the graph on two vertices and n edges joining them. Kinoshita [26] has proved the following nice result.

Theorem 4.1 (Kinoshita [26]). Any given n(n − 1)/2 knot types are realized by a spatial embedding of n at once.

Fig. 5 (obtained from [61, p. 208]) illustrates a 3 that realizes the trefoil, the ﬁgure-eight and the torus knot T(5, 2) simultaneously. Theorem 4.1 naturally yields to consider whether a given collection of knot types is realized by a spatial graph at once. Let (G) be the set of all cycles in a graph G and SE(G) the set of all spatial embeddings of G into the 3-sphere. Suppose that for each ∈ (G), ∈ SE() is given. We say that a set of embeddings { ∈ SE() | ∈ (G)} is realizable if there is an element f ∈ SE(G) such that the restriction map f | is ambient isotopic to for all ∈ (G). A graph G is adaptable if any set of embeddings { ∈ SE() | ∈ (G)} is realizable. Harikae and Kinoshita [24] proved a stronger form of Kinoshita’s result and used it to show that the graph consisting of three vertices and prescribed number of edges joining the three pairs of vertices is also adaptable. Yamamoto [66] showed that K4 is adaptable. J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 233

Trefoil Figure-eight T(5,2)

Fig. 5. A 3 embedding.

Theorem 4.2 (Yamamoto [66]). Let (c1,...,c7) be the seven cycles in K4. For any ordered 7-tuple (k1,...,k7) of knot types, there is a R(K4) such that the associated list of knot types of (c1,...,c7) is (k1,...,k7).

In [68], Yasuhara gives a certain canonical representation of knots that is obtained from -unknotting operation and a sufﬁcient condition for graphs to be adaptable by using this canonical representation. As an application, Yasuhara obtained the following result.

Theorem 4.3 (Yasuhara [68]). Any proper subgraphs of K5 and wheel graphs are adaptable.

Ohyama and Tanimaya [43] studied the Vassiliev invariants of the knots contained in a R(G) and obtained, among other results, that

Theorem 4.4 (Ohyama and Taniyama [43]). The graph C5, that is, the graph obtained from a 5-cycle by doubling all its edges (see graph G8 in Fig. 6) is not adaptable.

It is noted in [43] that G5 is the first planar graph which is known to be nonadaptable. Note that if H is a proper minor of G and G is adaptable, then H is also adaptable. Let be the set of nonadaptable graphs all of whose proper minors are adaptable. By Theorem 2.3, has a finite number of elements. It would be an interesting problem to find all elements of . Let 0 be the set of planar graphs in .

Theorem 4.5 (Yasuhara [68]). \ 0 ={K5, K3,3}.

The fact that K5 and K3,3 are not adaptable was shown by Motohashi and Taniyama [33].

Theorem 4.6 (Motohashi and Taniyama [33]). No nonplanar graph is adaptable.

Recently, Taniyama and Yasuhara [64] (see also [61]) proved that 234 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

GGGG 1 2 3 4

G5 G6 G7 G8

Fig. 6. Graphs G1,...,G8.

Theorem 4.7 (Taniyama and Yasuhara [64]). Let G1,...,G8 be the graphs shown in Fig. 6. Then, {G1,...,G8}⊂ 0.

Moreover, Taniyama and Yasuhara showed the following two results:

Theorem 4.8 (Taniyama and Yasuhara [64]). A set of spatial embeddings { ∈ SE() | ∈ (K5)} is realizable up to ambient isotopy if and only if there is an integer m such that m(m − 1) a ( ()) − a ( ()) = , 2 2 2 ∈C5(K5) ∈C4(K5) where a2(J ) is the second coefﬁcient of the Conway polynomial of knot J and C5(K5) (resp. C4(K5)) denote the set of all 5-cycles (resp.4-cycles) in K5.

Theorem 4.9 (Taniyama and Yasuhara [64]). A set of spatial embeddings { ∈ SE() | ∈ (K3,3)} is realizable up to ambient isotopy if and only if there is an integer m such that m(m − 1) a ( ()) − a ( ()) = , 2 2 2 ∈C6(K3,3) ∈C4(K3,3) where a2(J ) is the second coefﬁcient of the Conway polynomial of knot J and C6(K3,3) (resp. C4(K3,3)) denote the set of all 6-cycles (resp.4-cycles) in K3,3.

Since {m(m − 1)/2|m ∈ Z} is a proper subset of Z, then K5 and K3,3 are not adaptable. Thus, by Kuratowski graph planary criteria, Theorems 4.8 and 4.9 imply Theorem 4.6. Onda [44] gave a necessary and sufﬁcient condition for the set { ∈ SE() | ∈ (G)} to be realizable where G is any of the graphs G1,...,G7 given in Fig. 6 (the problem for G8 still open). J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 235

In [27], Kinoshita proved that, given a disjoint embedding of m(m−1)/2 bipartite graphs K2,n, there is a R(Km,n) whose subgraphs K2,n are equivalent to the given ones. Taniyama [62] gave conditions for a collection of knot types of subgraphs of a graph to be realized by a spatial embedding of the whole graph at once (implying Kinoshita’s result). Finally, we mention that a higher dimensional analogue of adaptability is studied in [67].

5. Linear embeddings

Theorems 2.1 and 3.1 present an unavoidable phenomenon in sufficiently large complete graphs, which can be regarded as a Ramsey-type theorem. This raises the following natural question:Does any R(Kn) with sufficiently large n contain a prescribed knot type or link type? The answer to this question is negative in general (if one makes a local knot on each edge of R(Kn), then any cycle will involve a number of such local knots, which restricts the knot and link types). We may restrict spatial embeddings of graphs so as to forbid such phenomenon, for instance, when R(G) is linear, that is, if each edge of R(G) is represented by a straight line segment. Let r =r(L) be the smallest positive integer such that every linear R(Kn), nr, contains a set of cycles isotopic to link L. Negami [38] showed that for any given knot, link or spatial graph R(G) there is a sufficiently large complete graph Kn such that every linear R(Kn) always contains a subdivision of R(G). In particular, we have

Theorem 5.1 (Negami [38]). Let K be a knot. Then, there exists a ﬁnite number r = r(K) such that any linear R(Kn), nr, contain a cycle isotopic to K.

The strategy to prove Theorem 5.1 relies on a Ramsey-type result in connection to the cyclic polytope. The cyclic polytope of dimension d with n vertices Cd (n) was discovered by Carathéodory [12,13] and many times rediscovered; it is usually deﬁned as the convex hull d in R , d 2, of n, nd + 1, different points x(t1),...,x(tn) of the moment curve x : R → Rd ,t→ (t, t2,...,td ). Cyclic polytopes, and simplicial neighborly polytopes, in general, play an important role in the combinatorial convex geometry due to their connection with certain extremal problems. For example, the upper bound theorem established by McMullen [31,32], says that the number of j-dimensional faces of a k-polytope with n vertices is maximal for Cd (n). A classical Ramsey theorem states

Theorem 5.2 (Ramsey [49]). Given any two integers r n, there exists an integer m such Em V that for every 2-colouring of ( r ) there is an n-element subset V of Em such that ( r ) is a Em monochromatic subset of ( r ).

The following proposition can be easily obtained from Theorem 5.2.

Proposition 5.3 (Björner et al. [3, Proposition 9.4.7]). Let n, d be integers with nd + 13. There exists an integer m=m(n, d) such that every set of m points in general position in afﬁne d-space contains the m vertices of a cyclic polytope of dimension d. 236 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

Proof of Theorem 5.1 (sketch). It is shown that, given a knot K, there always exists a cycle C made of line segments, with extremes at the set of m vertices of a cyclic polytope of dimension 3 with m sufﬁciently large such that C is ambient isotopic to K. And thus, by Proposition 5.3, the integer r is ﬁnite.

An older application of Ramsey theorem to knot theory can be found in [35], where Motzkin showed the existence of a bound R such that every general position set with at least R points contains a knotted polygon. Motzkin discussed the existence of a certain special position in R3, using Ramsey’s theorem, and found an octahedral position in a sufﬁciently large general position set. It is clear that any octahedral position contains a knotted hexagon, which is isotopic to the trefoil. It is clear that r(L)s(L) where s(L) is the sticky number of link L. From Theorem 2.1 r( 2) = s( 2) = we have that 21 6 which is the best possible since 21 6. This can also be shown by using the geometric arguments given by Robinson in [57]. As we remarked above, Theorem 3.1 says nothing about the knot types contained in R(K7). Therefore, one may wonder whether in any linear R(K7) there is always a cycle equivalent to the trefoil (that is, ambient isotopic to either the trefoil or its mirror; see Appendix). The latter was shown to be true by Brown in [10] (unpublished). In [46], we also gave a proof of this fact.

Theorem 5.4 (Brown [10], Ramírez Alfonsín [46]). Any linear R(K7) contains a cycle isotopic to either the trefoil or its mirror.

Let us briefly describe the idea used in [46] to prove Theorem 5.4 that uses the machinery of the oriented matroids theory.3 It is well known that there is a natural way to associate an oriented matroid (a set of circuits) to a given configuration of points in the space. Thus, in order to prove Theorem 5.4, we proceed as follows. Let M(R(K7)) be the oriented matroid associated to the set of points of a linear R(K7). We proposed three sets of conditions (in terms of circuits) and showed that if M(R(K7)) satisfies at least one of these conditions, then R(K7) is forced to contain a cycle isotopic to either the trefoil or its mirror.In[46], we also showed that there exists a linear R(K6) in which all cycles are trivial knots and found that there exists a linear R(K7) containing either the trefoil or its mirror (but not both at the same time). Motivated by this new approach, we continued similar investigations by considering a little more complicated knots and links.

( ). r( 2)> Theorem 5.5 Ramírez and Alfonsín [47] 41 7.

Theorem 5.6 (Ramírez Alfonsín [48]). r(ﬁgure-eight) and r(T(5, 2))9.

s( 2) = s( ) = s(T ( , )) Notice that 41 ﬁgure-eight 7 and 5 2 7. In order to prove Theorem R( ) 2 5.5 (resp. Theorem 5.6) we gave a linear K7 not containing cycles isotopic to 41 (resp. isotopic to the ﬁgure-eight nor to T(5, 2)). In both cases, we used the vertices of the cyclic

3 The theory of oriented matroids provides a broad setting to describe geometric configurations. This theory considers the structure of dependencies in vector spaces over ordered fields; see [3]. J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 237 polytope and its combinatorial description in terms of oriented matroids (the well-known alternating oriented matroids). Negami’s proof of Theorem 5.1 uses the fact that there is a sufficiently large integer m such that the cyclic polytope C3(m) contains a cycle isotopic to a given knot. Unfortunately, Negami’s method does not give an order of magnitude of m. Let us see how Nagami proceeded. A given knot diagram D(K) is transformed into what Negami called a n-plat representation which is more well known as a n-braid representation (which certainly always exists by Alexander’s Theorem; see [1]). Such representation is oriented and must have only positive crossings. So, each negative crossings is changed into (2n + 1)(n − 1) crossings of the opposite sign obtaining a positive plat representation of K. Finally, Negami constructed a linear embedding of a positive plat representation on C3(m) inductively. The induction is long, involved and assumes ‘certain conditions’ that may increase the value of m which depend on the number of crossings of the positive plat representation. So, the magnitude of m heavily depends on the positive plat representation and thus is difficult to estimate. In [48], we propose a completely new approach (avoiding the above drawbacks) to show that any knot has a cyclic embedding. Our method is constructive and yields the following result.

Theorem 5.7 (Ramírez Alfonsín [48]). Let K be knot and let cr(K) be its crossing number. Then, there exists a cycle in C3(7cr(K)) isotopic to K.

Let f (n) be the smallest integer such that any set of at least f (n) points in R3, in general position, induces the 3-dimensional cyclic polytope on n vertices. Proposition 5.3 ensures the existence of f (n) and any explicit upper bound of f (n) would imply, by Theorem 5.7, an explicit upper bound for r(K). In [30], Miyauchi generalized Negami’s result for complete multipartite graphs.

Theorem 5.8 (Miyauchi [30]). For any given knot K there exist a pair of integers M,N such that any linear R(Km,n) with nN and mM contains a cycle isotopic to K.

In [30], Miyauchi conjectured that any R(K8) and any R(K5,5) (not necessarily linear) contains a cycle isotopic to the trefoil. A result closely related to Theorem 5.8 is due to Negami [39] in which the rectilinear projections are introduced. A rectilinear projection of a R(G) is a projection consisting of only straight edges (note that this type of projections also exclude local knots on edges); see also [40].

Theorem 5.9 (Negami [39]). For any given knot K there exist a pair of integers M,N such that any R(Km,n) whose projection is rectilinear with nN and mM contains a cycle isotopic to K.

In [41], Negami showed that Ramsey theorem for spatial graphs without local knots does not hold in general. Negami constructed a R(Kn,n) which has no local knots on edges and which contains any subdivision of a given nonsplittable 2-component link. 238 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

III

Fig. 7. Reidemeister moves.

Appendix

A link L with k components consists of k disjoint simple closed curves in R3.Aknot K is a link with one component. A knot K is said to be trivial if K bounds a 2-cell in R3 and equivalently if there is a homeomorphism h : R3 → R3 which carries K onto the unit circle in the xy-plane {(x, y, 0) ∈ R3|x,y ∈ R}. Also a link L is said to be trivial if there is a homeomorphism h : R3 → R3 such that h(L) is contained in the xy-plane. In other words, a trivial link is a union of mutually unlinked trivial knots. Two knots (or links) K1 and K2 are said to be equivalent or to have the same knot type 3 3 (or link type) if there is a homeomorphism h : R → R such that h(K1) = K2.Anambient isotopy of R3 is a continuous map H : R3 × I → R3 such that Ht is a homeomorphism for any t ∈ I and H0 is the identity map, where I stands for the unit interval [0, 1] and 3 3 Ht : R → R is a map deﬁned by Ht (x) = H(x,t). Two links L1 and L2 are said to be 3 3 ambient isotopic if there is an ambient isotopy H : R × I → R such that H1(L1) = L2. A link diagram D(L) is obtained from L by projecting it into a plane in such a way that the projection of each component is smooth and at most two curves (not necessarily representing two different components) intersect (transversally) at any point. At each crossing point of the link diagram the curve which goes over the other is speciﬁed. The fundamental theorem of Reidemeister [50] states.

Theorem A.1 (Reidemeister [50]). Two links L1 and L2 are ambient isotopic if and only if any link diagram D(L1) can be transformed into any link diagram D(L2) by a ﬁnite sequence of moves I, II and III and their inverses; see Fig. 7.

A link L is said to be split if there is an embedding of a 2-sphere S2 in R3 − L such that each component of R3 − S2 contains a component of L. The mirror of a link L, denoted by L∗, is obtained by a reﬂection of L in a plane.A knot K is amphicheiral if K is isotopic to K∗. It is well known that the trefoil and its mirror are not isotopic; that is there is no sequence J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 239

Link or KnotDiagram Stick representation Sticky number

Trefoil 6

Figure-eight 7

T(5,2) 7

Hopf link 2 6 2 1

2 7 41

Fig. 8. Some basic knots and links.

of Reidemeister moves which will transform the trefoil to its mirror. In other words, the 2 trefoil is not amphicheiral; see [17]. It is also known that the Hopf link, denoted by 21 (the 2 simplest nontrivial link) is amphicheiral whilst the link 41 (the second simplest nontrivial link) is not; see [36,37]. In Fig. 8, a diagram of the simplest nontrivial knots and links as well as a sticky representation together with their sticky number (the sticky number, s(L) of a link L is the smallest number of sticks needed to represent L in R3) are given. We refer the reader to [11,37] for further deﬁnitions and terminology on knot theory. 240 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

References

[1] J.W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923) 93–95. [2] R. Bacher, Y. Colin de Verdière, Multiplicitités des valeurs propres et transformations étoile-triangle des graphes, Bull. la Société Mathématique France 123 (4) (1995) 517–533. [3] B. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids, Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1993. [4] T. Bohme, ˝ On spatial representations of graphs, Contemporary Methods in Graph Theory, Bibliograph. Inst., Mannheim (1990) 151–167. [5] T. Bohme, ˝ H. Sachs, Über räumliche Darstellungen endlicher Graphen (German) [On spatial representations of finite graphs], in 2nd Colloquium for Geometry and Combinatorics 1 and 2 Tech. Hochschule Karl-Marx- Stadt, Karl-Marx-Stadt (1983) 29–42. [6] T. Bohme, ˝ H. Sachs, Über räumliche Darstellungen endlicher Graphen (German) [On spatial representations of finite graphs], Mitt. Math. Ges. DDR 1 (1983) 5–31. [7] D. Bokor, B. Johnson, Links in spatial embeddings of Kn,n and Kn,n,1, preprint. [8] J.A. Bondy, U.S.R. Murty, Graph Theory and its Applications, MacMillan, New York, 1970. [9] H.-G. Bothe, Problem P 855, Colloq. Math. 28 (1973) 163; H.-G. Bothe, New Scottish Book, Problem 876 (1972) . [10] A.F. Brown, Embeddings of graphs in E3, Ph.D. Dissertation, Kent State University, 1977. [11] G. Burden, A. Zieschang, Knots, Studies in Mathematics, vol. 15, De Gruyter, New York, 1985. [12] C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potentzreihen die gegeberne Wete nicht annehemen, Math. Ann. 64 (1904) 95–115. [13] C. Carathéodory, Über den Variabilitätsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen, Rendiconto del Circolo Matematico di Palermo 32 (1911) 193–217; reprinted in:H. Tietze (Ed.), Constantin Carathéodory, Gesammelte Mathematische Schriften, Band III, C.H. Beck’sche Verlagsbuchhandlung, Münichen, 1955, pp. 78–110. [14] Y. Colin de Verdière, Sur un nouvel invariant des graphes et un critère de planarité, J. Combin. Theory Ser. B 50 (1990) 11–21. [15] Y.Colin deVerdière,On a new graph invariant and a criterion for planarity, Graph structure theory (Seattle,WA, 1991), in:N. Robertson, P. Seymour (Eds.), Contemporary Mathematics, vol. 147, American Mathematical Society, Providence, RI, 1993, pp. 137–147. [16] J.H. Conway, C.McA. Gordon, Knots on spatial graphs, J. Graph Theory 7 (1983) 445–453. [17] M. Dehn, Die beiden Kleeblattschlingen, Math. Ann. 69 (1914) 402–413. [18] A. Dochtermann, S. Chan, E. Kunz, T. Lalonde, Q. Loney, Some links properties of spatially embedded graphs, preprint. [19] E. Flapan, Intrinsic knotting and linking of complete graphs, Algebraic Geometric Topology 2 (2002) 371–380. [20] E. Flapan, J. Foisy, R. Naimi, J. Pommersheim, Intrinsically n-linked graphs, J. Knot Theory Ramifications 10 (8) (2001) 1143–1154. [21] E. Flapan, R. Naimi, J. Pommersheim, Intrinsically triple linked complete graphs, Topology Appl. 115 (2) (2001) 239–246. [22] J.S. Foisy, Intrinsically knotted graphs, J. Graph Theory 39 (2002) 178–187. [23] S. Chan, A. Dochtermann, J. Foisy, J. Hespen, E. Kunz, T. Lalonde, Q. Loney, K. Sharrow, N. Thomas, Graphs with disjoint links in every spatial embedding. J. Knot Theory Ramifications 13 (6) (2004) 737–748. [24] T. Harikae, S. Kinoshita, On spatial embeddings of R3-graphs, Kwansei Gakuin Univ.Annual Stud. 40 (1991) 145–148. [25] B. Johnson, W. Johnson, On the size of links in Kn,n,Kn,n,1 and Kn, J. Knot Theory Ramifications 11 (2) (2002) 145–151. [26] S. Kinoshita, On n-curves in R3 and their constituent knots, in:S. Suzuki (Ed.), Topology and Computer Science, Proceedings of the Symposium Held in Honour of Shin’ichi Kinoshita, Hiroshi Noguchi and Tatsu Homma on the Occasion of their Sixtieth Birthday, Kinokuniyama, Tokyo, Japan, 1987, pp. 211–216. [27] S. Kinoshita, On spatial bipartite Km,n’s and their constituent K2,n’s, Kobe J. Math. 8 (1991) 41–46. [28] T. Kohara, S. Suzuki, Some Remarks on Knots and Links in Spatial Graphs, Knots 90, Osaka, 1990, De Gruyter (Walter), Berlin, New York, 1992, pp. 435–445. J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242 241

[29] L. Lovász, A. Schrijver, A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs, Proc. Amer. Math. Soc. 126 (1998) 1275–1285. [30] M.S. Mayauchi, Topological Ramsey theorem for complete bipartite graphs, J. Combinat. Theory Ser. B 62 (1994) 164–179. [31] P. McMullen, The maximal number of faces of a convex polytope, Mathematika 17 (1970) 179–184. [32] P. McMullen, G.C. Shephard, Convex Polytopes and the Upper Bound Conjecture, London Mathematical Society Lecture Note Series 3, Cambridge University Press, Cambridge, MA, 1971. [33] T. Motohashi, K. Taniyama, Delta-unknotting operation and vertex homotopy of graphs in R3, in:S. Suzuki (Ed.), Proceedings of KNOTS ’96, Tokyo, World Science Publishing, River Edge, NJ, 1997, pp. 185–200. [34] R. Motwani, A. Raghunathan, H. Saran, Constructive results from graph minors:linkless embeddings, 29th Annual Symposium on Foundations of Computer Science, IEEE, 1998, pp. 389–409. [35] T.H. Motzkin, Cooperative classes of finite sets in one and more dimensions, J. Combinat. Theory 3 (1967) 244–251. [36] K. Murasugi, On the signature of links, Topology 9 (1970) 283–298. [37] K. Murasugi, Knot Theory and its Applications, Birkhaüser, Basel, 1996. [38] S. Negami, Ramsey theorems for knots, links and spatial graphs, Trans. Amer. Math. Soc. 324 (1991) 527–541. [39] S. Negami, Ramsey-type theorem for spatial graphs, Graphs Combin. 14 (1998) 75–80. [40] S. Negami, Ramsey-type theorems for spatial graphs and good drawings, J. Combinat. Theory Ser. B 72 (1998) 53–62. [41] S. Negami, Note on Ramsey theorems for spatial graphs, Theoret. Comput. Sci. 263 (2001) 205–210. [42] J. Nešetˇril, R. Thomas, A note on spatial representation of graphs, Comment. Math. Univ. Carolin. 26 (1985) 655–659. [43] Y. Ohyama, K. Taniyama, Vassiliev invariants of knots in a spatial graph, Pacific J. Math. 200 (1) (2001) 191–205. [44] K. Onda, A characterization of knots in a spatial graph, J. Knot Theory Ramifications 9 (2000) 1069–1084. [45] T. Otsuki, Knots and links in certain spatial complete graphs, J. Combinat. Theory Ser. B 68 (1996) 23–35. [46] J.L. Ramírez Alfonsín, Spatial graphs and oriented matroids:the Trefoil, Discrete Comput. Geom. 22 (1999) 149–158. [47] J.L. Ramírez Alfonsín, On linked spatial representations, J. Knot Theory Ramifications 10 (2001) 143–150. [48] J.L. Ramírez Alfonsín, Spatial graphs, knots and the cyclic polytope, submitted. [49] F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930) 264–286. [50] K. Reidemeister, Homotopieringe und Linsenräume, Sem. Hamburg 11 (1936) 102–109. [51] N. Robertson, P. D. Seymour, Graph minors. XX. Wagner’s conjecture, J. Combin. Theory Ser. B 92 (2) (2004) 325–357. [52] N. Robertson, P. Seymour, R. Thomas, A survey of linkless embeddings, Graph structure theory, Seattle, WA, 1991, in:N. Robertson, P. Seymour (Eds.), Contemporary Mathematics, vol. 147, American Mathematical Society, Providence, RI, 1993, pp. 125–136. [53] N. Robertson, P. Seymour, R. Thomas, Linkless embeddings of graphs in 3-space, Bull. Amer. Math. Soc. 28 (1993) 84–89. [54] N. Robertson, P. Seymour, R. Thomas, Kuratowski chains, J. Combinat. Theory Ser. B 64 (1995) 127–154. [55] N. Robertson, P. Seymour, R. Thomas, Petersen family minors, J. Combinat. Theory Ser. B 64 (1995) 155–184. [56] N. Robertson, P. Seymour, R. Thomas, Sachs’ linkless embedding conjecture, J. Combinat. Theory Ser. B 64 (1995) 185–227. [57] D.F. Robinson, Symmetric embeddings of graphs, J. Combinat. Theory 9 (1970) 377–400. [58] H. Sachs, On spatial representation of finite graphs, Finite and Infinite Sets, Colloq. Math. Soc. Janos Bolyai, vol. 37, North-Holland, Amsterdam, 1981. [59] H. Sachs, On a spatial analogue of Kuratowski’s theorem on planar graphs—an open problem, Graph theory, in:M. Borowiecki, J.W. Kennedy, M.M. Syslo (Eds.), Proceedings of the Conference in Memoriam K. Kuratowski, Lagow, Poland, 1981, Lecture Notes in Mathematics, vol. 1018, Springer, Berlin, Heidelberg, 1983, pp. 230–241. [60] M. Shimabara, Knots in certain spatial graphs, Tokyo J. Math. 11 (2) (1988) 405–413. [61] K. Taniyama, On spatial graphs, in:S. Suzuki (Ed.), Proceedings of KNOTS ’96, World Science Publishing, River Edge, NJ, 1997, pp. 195–217. 242 J.L. Ramírez Alfonsín / Discrete Mathematics 302 (2005) 225–242

[62] K. Taniyama, Knotted subgraphs in a spatial graph, Kobe J. Math. 14 (1997) 207–212. [63] K. Taniyama, Higher dimensional links in a simplicial complex embedded in a sphere, Paciﬁc J. Math. 194 (2) (2000) 465–467. [64] K. Taniyama, M. Yasuhara, Realization of knots and links in a spatial graph, Topology Appl. 112 (2001) 87 –109. [65] R. Thomas, Recent excluded minor theorems for graphs, in:J.D. Lamb, D.A. Preece (Eds.), Surveys in Combinatorics, 1999, London Mathematical Society, Lecture Notes Series, vol. 267, Cambridge University Press, Cambridge, 1999, pp. 201–222. [66] M.Yamamoto, Knots in spatial embeddings of the complete graph on four vertices, TopologyAppl. 36 (1990) 291–298. [67] A. Yasuhara, On higher n-dimensional curves, Kobe J. Math. 8 (1991) 191–196. [68] A. Yasuhara, Delta-unknotting operation and adaptability of certain graphs, in:S. Suzuki (Ed.), Proceedings of KNOTS ’96, Tokyo, World Science Publishing, River Edge, NJ, 1997, 115–121.