A refinement of the Conway-Gordon theorems

A refinement of the Conway-Gordon theorems

Abstract.

In 1983, Conway-Gordon showed that for every spatial complete graph on vertices, the sum of the linking numbers over all of the constituent -component links is congruent to modulo , and for every spatial complete graph on vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to modulo . In this article, we give integral lifts of the Conway-Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown-Ramírez Alfonsín and Huh-Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods.

Key words and phrases:
Spatial graph, Conway-Gordon theorem, Rectilinear spatial graph
1991 Mathematics Subject Classification:
Primary 57M15; Secondary 57M25
The author was partially supported by Grant-in-Aid for Young Scientists (B) (No. 21740046), Japan Society for the Promotion of Science.

1. Introduction

Throughout this paper we work in the piecewise linear category. Let be an embedding of a finite graph into . Then (or ) is called a spatial embedding of or simply a spatial graph. Two spatial embeddings and of are said to be ambient isotopic if there exists an orientation-preserving self homeomorphism on such that . We call a subgraph of which is homeomorphic to the circle a cycle of , and a cycle of which contains exactly edges a -cycle of . We denote the set of all cycles of , the set of all -cycles of and the set of all pairs of two disjoint cycles consisting of a -cycle and an -cycle of by , and , respectively. For (resp. ) and a spatial embedding of , (resp. ) is none other than a knot (resp. -component link) in .

Let be the complete graph on vertices, namely the graph consisting of vertices , a pair of whose vertices and is connected by exactly one edge if . Let us recall the following two famous theorems in spatial graph theory, which are called the Conway-Gordon theorems.

Theorem 1-1.

(Conway-Gordon [4]) For any spatial embedding of , it follows that

(1.1)

where denotes the linking number in . In particular, for any spatial embedding of , there exists a pair of two disjoint -cycles of such that is a non-splittable -component link with odd linking number.

Theorem 1-2.

(Conway-Gordon [4]) For any spatial embedding of , it follows that

(1.2)

where denotes the Arf invariant [16]. In particular, for any spatial embedding of , there exists a -cycle of such that is a non-trivial knot with Arf invariant one.

Conway-Gordon’s formulas (1.1) and (1.2) hold only modulo two. Note that the square of the linking number is congruent to the linking number modulo two, and the second coefficient of the Conway polynomial of a knot is congruent to the Arf invariant modulo two [7]. Our first purpose in this article is to refine Conway-Gordon’s formulas above by giving their integral lifts in terms of the square of the linking number and the second coefficient of the Conway polynomial as follows.

Theorem 1-3.

For any spatial embedding of , we have that

where denotes the second coefficient of the Conway polynomial.

Theorem 1-4.

For any spatial embedding of , we have that

Note that Theorems 1-1 and 1-2 can be obtained from Theorems 1-3 and 1-4 respectively by taking the modulo two reduction. We also show that Theorem 1-4 can be divided into the following two formulas.

Corollary 1-5.

For any spatial embedding of , we have that

Note that Theorem 1-4 can be recovered from (1-5) and (1-5). We give proofs of Theorems 1-3, 1-4 and Corollary 1-5 in section . We remark here that the second coefficient of the Conway polynomial and the square of the linking number are Vassiliev invariants [21] of order [2], [10]. See also [14] for general results about Vassiliev invariants of knots and links in a spatial graph.

Our second purpose in this article is to give applications of Theorems 1-3 and 1-4 to the theory of rectilinear spatial graphs. Here a spatial embedding of a graph is said to be rectilinear if for any edge of , is a straight line segment in . Note that for any positive integer , there exists a rectilinear spatial embedding of . Actually it can be constructed by taking vertices on the moment curve in and connecting every pair of two distinct vertices by a straight line segment (this also implies that any simple graph has a rectilinear spatial embedding). Note that every knot or link contained in a rectilinear embedding of has stick number less than or equal to , where the stick number of a link (or a knot) is the minimum number of edges in a polygon which represents . The following are fundamental results on stick numbers for knots and -component links, see [11], [1], [8].

Proposition 1-6.

(1) For any non-trivial knot , it follows that . Moreover, if and only if is a trefoil knot, and if and only if is a figure eight knot.

(2) For a -component link , it follows that . Moreover, if and only if is either a trivial link or a Hopf link, and if and only if is a -torus link.

Therefore we have that non-trivial knots in a rectilinear spatial embedding of (resp. ) are only trefoil knots (resp. trefoil knots and figure eight knots), and non-trivial -component links in a rectilinear spatial embedding of (resp. ) are only Hopf links (resp. Hopf links and -torus links). Then, as a result for rectilinear spatial embeddings of corresponding to Theorem 1-2, the following is known.

Theorem 1-7.

(Brown [3], Ramírez Alfonsín [15]) For any rectilinear spatial embedding of , contains a trefoil knot.

We remark here that Brown’s paper [3] is unpublished, and Ramírez Alfonsín’s proof in [15] is done by applying oriented matroid theory with the help of a computer. We refine Theorem 1-7 by an application of Theorems 1-3 and 1-4 as follows.

Theorem 1-8.

For any rectilinear spatial embedding of , is a positive odd integer. In particular, if and only if the non-trivial -component links in are exactly twenty one Hopf links.

By Theorem 1-8, for any rectilinear spatial embedding of , there exists a -cycle of such that . Since , by Proposition 1-6 (1), must be a trefoil knot because and . Namely Theorem 1-7 is obtained from Theorem 1-8 as a corollary. We give a proof of Theorem 1-8 in section . Note that our proof is topological whereas Ramírez Alfonsín’s proof is extremely combinatorial and computational.

On the other hand, as a result for rectilinear spatial embeddings of corresponding to Theorem 1-1, the following is known.

Theorem 1-9.

(Huh-Jeon [6]) For any rectilinear spatial embedding of , contains at most one trefoil knot and at most three Hopf links. In particular,

(1) does not contain a trefoil knot if and only if contains exactly one Hopf link.

(2) contains a trefoil knot if and only if contains exactly three Hopf links.

Although Huh and Jeon do not use a computer in [6], their proof is in the same spirit as Ramírez Alfonsín’s in that it is also purely combinatorial. As an application of Theorem 1-3, we give an alternative topological proof of Theorem 1-9 in section 4.

2. Spatial graph-homology invariants

In this section, we introduce some homological invariants of spatial graphs which are needed later. Let be or , where denotes the complete bipartite graph on vertices, namely the graph consisting of vertices a pair of whose vertices and are connected by exactly one edge if is odd. We give an orientation to each edge of as illustrated in Fig. 2.1. For an unordered pair of disjoint edges of , we define the sign by , and . For an unordered pair of disjoint edges of , we also define the sign by , and if and are parallel in Fig. 2.1 and if and are anti-parallel in Fig. 2.1. For a spatial embedding of , we fix a regular diagram of and denote the sum of the signs of all crossing points between and by , where is an unordered pair of disjoint edges of . Then, an integer defined by

where the sum is taken over all unordered pairs of disjoint edges of , is called the Simon invariant of .

Figure 2.1.

It is known that is an odd integer valued ambient isotopy invariant of [19]. Moreover, the following is known.

Proposition 2-1.

([19]) Let be a spatial embedding of or . Then is a spatial graph-homology invariant of .

Here, a spatial graph-homology is an equivalence relation on spatial graphs introduced in [19] as a generalization of a link-homology on oriented links. We refer the reader to [19] for the precise definition of a spatial graph-homology.

On the other hand, let be , or , where is the graph as illustrated in Fig. 2.2. Let be a map defined by

if ,

if and

if . For a spatial embedding of , an integer defined by

is called the -invariant of [18]. Then the following holds.

Proposition 2-2.

(1) (Motohashi-Taniyama [9]) Let be a spatial embedding of or . Then it follows that

(2) (Taniyama-Yasuhara [20]) Let be a spatial embedding of . We denote the pair of disjoint two -cycles and resp. and of by resp. Then it follows that

Figure 2.2.

3. Proofs of Theorems 1-3 and 1-4

First we give a proof of Theorem 1-3. We can see that there exist exactly ten subgraphs of each of which is isomorphic to , exactly six subgraphs of each of which is isomorphic to and exactly ten pairs of disjoint -cycles of . Then we have the following.

Lemma 3-1.

Let be a spatial embedding of . Then we have that

Proof.

For any spatial embedding of , there exist ten integers and such that is spatial graph-homologous to the spatial embedding of as illustrated in Fig. 3.1 [17],[12], where the rectangle represented by an integer stands for half twists as illustrated in Fig. 3.2. In general, for two spatial embeddings and of a graph which are spatial graph-homologous, and are also spatial graph-homologous for any subgraph of the graph by the definition. We recall that the Simon invariant is a spatial graph-homology invariant (Proposition 2-1), and the linking number is also a typical spatial graph-homology invariant. Thus we have that

Therefore it is sufficient to show that

(3.1)
Figure 3.1.
Figure 3.2.

We may assume that and are spatial graphs as illustrated in Fig. 3.3 (1) and (2) respectively, and are spatial graphs as illustrated in Fig. 3.4 (1) and (2) respectively, and and are -component links as illustrated in Fig. 3.5 (1) and (2) respectively, where we regard and for .

Figure 3.3.
Figure 3.4.
Figure 3.5.

Note that the square of the linking number is an invariant of a non-oriented -component link, and the square of the Simon invariant is also an invariant of non-labeled spatial embeddings of and (see [13, Lemma 3.2]). Thus we may calculate and by assigning the orientations of edges as illustrated in Fig. 3.3 and 3.4, respectively, and by assigning the orientations of components as illustrated in Fig. 3.5. Then we have that

Then by a direct calculation, it is not hard to see that (3.1) holds. ∎

Proof of Theorem 1-3..

By Proposition 2-2 (1), we have that

(3.2)
(3.3)

Thus by (3.2), (3.3) and Lemma 3-1, we have that

Then we can see that for any -cycle (resp. -cycle) of there exists exactly one (resp. ) such that is a -cycle of (resp. -cycle of ), and for any -cycle of there exist exactly two ’s (resp. ’s) such that is a -cycle of (resp. ). Thus we have that

(3.5)
(3.6)
(3.7)

Therefore by (3), (3.5), (3.6) and (3.7), we have the desired conclusion. ∎

Next we give a proof of Theorem 1-4. In the following we denote a path of length of consisting of two edges and by , and the subgraph of obtained from by deleting the vertex and all of the edges incident to by . Actually is isomorphic to for any .

Proof of Theorem 1-4..

For , let be the subgraph of obtained from by deleting the edges and . Note that is homeomorphic to , namely is obtained from the graph isomorphic to by subdividing an edge by the vertex , see Fig. 3.6.

Figure 3.6.

Let be a spatial embedding of . Then by applying Theorem 1-3 to , we have that

Let us take the sum of both sides of (3) over . For a -cycle of , let and be the two vertices of which are adjacent to in (). Then is a -cycle of . This implies that

(3.9)

For a -cycle of , let be an edge of which is not contained in . Then is a -cycle of which does not contain . Note that there are nine ways to choose such a pair of and . This implies that

(3.10)

For a -cycle of which contains the vertex , let and be the two vertices of which are adjacent to in . Then is a -cycle of which contains . This implies that

(3.11)

For a -cycle of , let be an edge of which is not contained in . Then is a -cycle of which does not contain . Note that there are ten ways to choose such a pair of and . This implies that

(3.12)

For a pair of disjoint cycles of consisting of a -cycle which contains the vertex and a -cycle , let and be the two vertices of which are adjacent to in . Then is a pair of disjoint cycles of consisting of a -cycle which contains and a -cycle . This implies that

(3.13)

For a pair of disjoint -cycles of , let be an edge of which is not contained in . Then is a pair of disjoint -cycles of which does not contain . Note that there are nine ways to choose such a pair of and . This implies that

(3.14)

Thus by (3), (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14), we have that

Then, by applying Theorem 1-3 to , we have that

By combining (3) and (3), we have that

Note that this also implies that

Now we take the sum of both sides of (3) over . For a -cycle of , let be a vertex of which is contained in . Note that there are six ways to choose such a vertex . This implies that

(3.18)

For a -cycle of , let be a vertex of which is not contained in . Then is a -cycle of . Note that there are two ways to choose such a vertex . This implies that

(3.19)

For a pair of disjoint cycles of which consisting of a -cycle and a -cycle , let be a vertex of which is contained in . Note that there are four ways to choose such a vertex . This implies that

(3.20)

Finally, by combining (3), (3.18), (3.19) and (3.20), we have the desired conclusion. ∎

To prove Corollary 1-5, we show the following lemma.

Lemma 3-2.

For any spatial embedding of