Generalizations of the Conway-Gordon theorems

# Generalizations of the Conway-Gordon theorems and intrinsic knotting on complete graphs

## Abstract.

In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway-Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway-Gordon theorems to complete graphs with arbitrary number of vertices greater than or equal to six. As an application, we show that for every rectilinear spatial complete graph on greater than or equal to six vertices, the sum of the second coefficients of the Conway polynomials over all of the Hamiltonian knots is determined explicitly in terms of the number of triangle-triangle Hopf links.

###### Key words and phrases:
Spatial graphs, Conway-Gordon theorems
###### 1991 Mathematics Subject Classification:
Primary 57M15; Secondary 57M25
The second author was supported by JSPS KAKENHI Grant Number 15K04881.

## 1. Introduction

Throughout this paper we work in the piecewise linear category. Let be a finite graph. An embedding of into the -dimensional Euclidean space is called a spatial embedding of , and the image is called a spatial graph of . Two spatial embeddings and of are said to be equivalent if there exists a self homeomorphism on such that . We call a subgraph of homeomorphic to the circle a cycle of , and a cycle of containing exactly edges a -cycle of . In particular, a -cycle is also called a Hamiltonian cycle if equals the number of all vertices of . We denote the set of all -cycles of by . Moreover, we denote the set of all pairs of two disjoint cycles of consisting of a -cycle and an -cycle by . For a cycle (resp. a pair of disjoint cycles ) and a spatial embedding of , (resp. ) is none other than a knot (resp. -component link) in . In particular for a Hamiltonian cycle of , we also call a Hamiltonian knot in .

Let be the complete graph on vertices, that is the graph consisting of vertices, a pair of whose distinct vertices is connected by exactly one edge. Then the following fact is well-known as the Conway-Gordon theorem.

###### Theorem 1-1.

(Conway-Gordon [8])

1. For any spatial embedding of , it follows that

 ∑λ∈Γ3,3(K6)lk(f(λ))≡1(mod2),

where denotes the linking number in .

2. For any spatial embedding of , it follows that

 ∑γ∈Γ7(K7)a2(f(γ))≡1(mod2),

where denotes the second coefficient of the Conway polynomial.

The second coefficient of the Conway polynomial of a knot is also congruent to the Arf invariant of the knot modulo two [21, Corollary 10.8]. Theorem 1-1 implies that is intrinsically linked, that is, every spatial graph of contains a nonsplittable -component link, and is intrinsically knotted, that is, every spatial graph of contains a nontrivial knot. The Conway-Gordon theorem made a beginning of the study of intrinsic linking and knotting of graphs and has motivated the many research of intrinsic properties of graphs (see for example [11, §2-6]). On the other hand, as far as the authors know, there have been little results about a generalization of the Conway-Gordon type congruences for complete graphs on eight or more vertices. Our purposes in this paper are to generalize the Conway-Gordon theorem for complete graphs with arbitrary number of vertices greater than or equal to six and investigate the behavior of the nontrivial Hamiltonian knots in a spatial complete graph. First of all, we recall an integral Conway-Gordon theorem for which was proven by the second author as follows.

###### Theorem 1-2.

(Nikkuni [24]) For any spatial embedding of , it follows that

 2∑γ∈Γ6(K6)a2(f(γ))−2∑γ∈Γ5(K6)a2(f(γ)) = ∑λ∈Γ3,3(K6)lk(f(λ))2−1.

Note that Theorem 1-1 (1) can be recovered by taking the modulo two reduction of (1-2), namely Theorem 1-2 is an integral lift of Theorem 1-1 (1). In [24], an integral lift of Theorem 1-1 (2) was also given (see Theorem 2-2 (1)). In this paper, we generalize Theorem 1-2 for complete graphs with arbitrary number of vertices greater than or equal to six as follows.

###### Theorem 1-3.

Let be an integer. For any spatial embedding of , it follows that

 ∑γ∈Γn(Kn)a2(f(γ))−(n−5)!∑γ∈Γ5(Kn)a2(f(γ)) = (n−5)!2{∑λ∈Γ3,3(Kn)lk(f(λ))2−(n−15)}.

Note that any pair of two disjoint -cycles of is not common for any two different subgraphs of isomorphic to . Then Theorem 1-1 (1) implies that is greater than or equal to the number of subgraphs of isomorphic to , that is equal to , and by a direct calculation we have

 (1.3) (n−5)!2{(n6)−(n−15)}=(n−5)(n−6)(n−1)!2⋅6!.

Thus by (1.3) and Theorem 1-3, we have the following.

###### Corollary 1-4.

Let be an integer. For any spatial embedding of , it follows that

 ∑γ∈Γn(Kn)a2(f(γ))−(n−5)!∑γ∈Γ5(Kn)a2(f(γ))≥(n−5)(n−6)(n−1)!2⋅6!.
###### Remark 1-5.

Endo-Otsuki introduced a certain special spatial embedding of , a canonical book presentation of [9], and Otsuki also showed that contains exactly Hopf links corresponding to a pair of two disjoint -cycles of if [26]. Thus the lower bound of Corollary 1-4 is sharp. In particular for any -cycle of , is a trivial knot. Thus it follows that for an integer ,

 ∑γ∈Γn(Kn)a2(fb(γ))=(n−5)(n−6)(n−1)!2⋅6!.

On the other hand, also note that the parity of coincides with the parity of . Moreover it can be seen that is odd if and only if , and is odd if and only if by an application of Lucas’s theorem for binomial coefficients (see [10] for example). Then by Theorem 1-3, we also have the following congruence, that is a generalization of Theorem 1-1 (2).

###### Corollary 1-6.

Let be an integer. For any spatial embedding of , it follows that

 ∑γ∈Γn(Kn)a2(f(γ)) ≡ ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−(n−5)!2(n−15)(n≡0(mod8))0(n≢0,7(mod8))(n−5)!2(n6)(n≡7(mod8))(mod(n−5)!).

In particular, by applying the case of with Corollary 1-6, we have

 ∑γ∈Γ7(K7)a2(f(γ))≡2!2(76)≡1(mod2),

that is, Theorem 1-1 (2).

###### Proof of Corollary 1-6.

For any two spatial embeddings and of , by Theorem 1-3, we have

 ∑γ∈Γn(Kn)a2(f(γ))−∑γ∈Γn(Kn)a2(g(γ)) ≡ (n−5)!2(∑λ∈Γ3,3(Kn)lk(f(λ))2−∑λ∈Γ3,3(Kn)lk(g(λ))2)(mod(n−5)!).

Since and have the same parity, that is equal to the parity of , by (1), it follows that

 (1.5) ∑γ∈Γn(Kn)a2(f(γ))≡∑γ∈Γn(Kn)a2(g(γ))(mod(n−5)!).

Note that there exists a spatial embedding of such that

 (1.6) ∑λ∈Γ3,3(Kn)lk(g(λ))2=(n6),

see Remark 1-5 or Remark 1-12. Thus by (1.5) and (1.6), it follows that

 (1.7) ∑γ∈Γn(Kn)a2(f(γ))≡(n−5)!2{(n6)−(n−15)}(mod(n−5)!)

for any spatial embedding of . If , since is even, by (1.7), we have

 ∑γ∈Γn(Kn)a2(f(γ))≡0(mod(n−5)!).

If , since is even and is odd, by (1.7), we have

 ∑γ∈Γn(Kn)a2(f(γ))≡−(n−5)!2(n−15)(mod(n−5)!).

If , since is odd and is even, by (1.7), we have

 ∑γ∈Γn(Kn)a2(f(γ))≡(n−5)!2(n6)(mod(n−5)!).

This completes the proof. ∎

###### Remark 1-7.

For any spatial embedding of , it were shown that by Foisy [13] and by Hirano [15]. These results imply that , and it can also be shown by applying the case of with Corollary 1-6:

 ∑γ∈Γ8(K8)a2(f(γ))≡−3!2(75)=−63≡3(mod6).

Hirano also showed that for any spatial embedding of if [15]. Corollary 1-6 also generalizes it markedly.

Theorem 1-3 (and Corollary 1-4) is also useful for investigating the behavior of the nontrivial Hamiltonian knots in rectilinear spatial complete graphs. Here, a spatial embedding of a graph is said to be rectilinear if for any edge of , is a straight line segment in . In particular, a special rectilinear spatial embedding of can be constructed by taking vertices of in order on the moment curve in and connecting every pair of two distinct vertices and by a straight line segment, see Fig. 1.1 for . We call a standard rectilinear spatial embedding of . A rectilinear spatial graph appears in polymer chemistry as a mathematical model for chemical compounds (see [3, §7], for example), and the range of rectilinear spatial graph types is much narrower than the general spatial graphs. So we are interested in the behavior of the nontrivial Hamiltonian knots in a rectilinear spatial complete graph. Note that every knot or link contained in a rectilinear spatial graph of has stick number less than or equal to , where the stick number of a link is the minimum number of edges in a polygon which represents . The following is a collection of fundamental results on stick numbers for knots and links (see Adams [3, §1.6], Negami [23, Theorem 6], Adams-Brennan-Greilsheimer-Woo [4, Theorem 2.1] and Calvo [7, Theorem 1]), where we denote each of knots and links appearing in the statement by using the label of it in Rolfsen’s table [30].

###### Proposition 1-8.

Let be a link. Then the followings hold.

1. If is a nontrivial knot, then .

2. if and only if is equivalent to , or .

3. if and only if is equivalent to or .

4. if and only if is equivalent to , , , , , a granny knot , a square knot , , or .

Proposition 1-8 (1) says that every polygonal knot with less than or equal to five sticks is trivial. Thus for rectilinear spatial complete graphs, by Theorem 1-3 we have the following immediately.

###### Theorem 1-9.

Let be an integer. For any rectilinear spatial embedding of , it follows that

 ∑γ∈Γn(Kn)a2(fr(γ))=(n−5)!2{∑λ∈Γ3,3(Kn)lk(fr(λ))2−(n−15)}.

As we saw in Proposition 1-8 (2), a -component link with stick number six is either a trivial link or a Hopf link. Thus for any rectilinear spatial embedding of , is greater than or equal to as we mentioned before, and also equal to the number of “triangle-triangle” Hopf links in . On the other hand, it is known that there is a strong restriction on the number of Hopf links in a rectilinear spatial graph of as follows.

###### Theorem 1-10.

(Hughes [19], Huh-Jeon [17], Nikkuni [24]) Every rectilinear spatial graph of contains at most three Hopf links.

Theorem 1-10 implies that is less than or equal to , and by a direct calculation we have

 (1.8) (n−5)!2{3(n6)−(n−15)}=3(n−2)(n−5)(n−1)!2⋅6!.

Thus by (1.3), (1.8) and Theorem 1-9, we have the following.

###### Corollary 1-11.

Let be an integer. For any rectilinear spatial embedding of , it follows that

 (n−5)(n−6)(n−1)!2⋅6!≤∑γ∈Γn(Kn)a2(fr(γ))≤3(n−2)(n−5)(n−1)!2⋅6!.
###### Remark 1-12.

For the standard rectilinear spatial embedding of () and a subgraph of isomorphic to , it can be easily seen that the embedding restricted on is equivalent to the standard rectlinear spatial embedding of . Since the standard rectilinear spatial graph of contains exactly one nonsplittable -component link which is a Hopf link, it follows that contains exactly triangle-triangle Hopf links. Thus the lower bound in Corollary 1-11 is sharp. But the authors strongly think that it cannot be expected at all that the upper bound is sharp if , see Example 3-2.

As an application of Corollary 1-11, let us actually count the number of nontrivial Hamiltonian knots in a rectilinear spatial complete graph. For every spatial embedding of (which does not need to be rectilinear), Hirano showed that there exist at least three nontrivial Hamiltonian knots with an odd value of in [16], and Foisy showed that there exist at least nontrivial Hamiltonian knots with an odd value of in if [5]. On the other hand, Corollary 1-11 makes us possible to evaluate the number of nontrivial Hamiltoian knots with a positive value of in a rectilinear spatial graph of as follows. Let us denote the crossing number of a knot by . Then it has been shown that by Calvo [7, Theorem 4], and also has been shown that by Polyak-Viro [27, Theorem 1.E]. By combining these results, we have the following upper bound of the value of for every polygonal knot with sticks.

###### Proposition 1-13.

Let be a knot whose stick number is less than or equal to . Then it follows that

 a2(K)≤((n−3)2(n−4)2(32.

By Corollary 1-11 and Proposition 1-13, we can evaluate the number of nontrivial Hamiltonian knots with a positive value of in a rectilinear spatial graph which makes exceed the lower bound.

###### Corollary 1-14.

Let be an integer. The minimum number of nontrivial Hamiltonian knots with a positive value of in every rectilinear spatial graph of is at least

where and denote the ceiling function and the floor function, respectively.

The concrete values of for are given in the following table. Note that in the case of , we can obtain a better evaluation than of the number of nontrivial Hamiltonian knots with a positive value of in every rectilinear spatial graph of (see Example 3-4 and Remark 3-5), and is more than Foisy’s lower bound of the minimum number of nontrivial Hamiltonian knots with an odd value of if .

We shall devote the section 2 to a proof of Theorem 1-3. In section , we give examples and present some problems.

## 2. Proof of Theorem1-3

We show some lemmas which are needed to prove Theorem 1-3.

###### Lemma 2-1.
1. Let be an integer. For any spatial embedding of , it follows that

 2∑γ∈Γ6(Kn)a2(f(γ))−2(n−5)∑γ∈Γ5(Kn)a2(f(γ)) = ∑λ∈Γ3,3(Kn)lk(f(λ))2−(n6).
2. Let be an integer. For any spatial embedding of , it follows that

 ∑λ∈Γ3,4(Kn)lk(f(λ))2=2(n−6)∑λ∈Γ3,3(Kn)lk(f(λ))2.
###### Proof of Lemma 2-1 (1).

Note that each -cycle of is common for exactly subgraphs isomorphic to if . Then by applying Theorem 1-2 to the embedding restricted on each of the subgraphs of isomorphic to and taking the sum of both sides of (1-2) over all of them, we have the result. ∎

In order to prove Lemma 2-1 (2), we recall integral Conway-Gordon type theorems for spatial embeddings of and which were proven by the second author [24] and O’Donnol [25], respectively. Here, the complete -partite graph is the graph whose vertex set can be decomposed into mutually disjoint nonempty sets where the number of elements in equals such that no two vertices in are connected by an edge and every pair of vertices in the distinct sets and is connected by exactly one edge. See Fig. 2.1 which illustrates and . In particular for , let us denote the subgraph of which is isomorphic to and does not contain by .

###### Theorem 2-2.
1. (Nikkuni [24]) For any spatial embedding of , it follows that

 7∑γ∈Γ7(K7)a2(f(γ))−6∑γ∈Γ6(K7)a2(f(γ))−2∑γ∈Γ5(K7)a2(f(γ)) = 2∑λ∈Γ3,4(K7)lk(f(λ))2−21.
2. (O’Donnol [25]) For any spatial embedding of , it follows that

 2∑γ∈Γ7(K3,3,1)a2(f(γ))−4∑γ∈Γ6(H)a2(f(γ))−2∑γ∈Γ5(K3,3,1)a2(f(γ)) =

Then by applying Theorem 2-2 (2) to each of the subgraphs of isomorphic to and combining with Theorem 2-2 (1), we also have the following equation for every spatial embedding of .

###### Theorem 2-3.

For any spatial embedding of , it follows that

 (2.1) ∑λ∈Γ3,4(K7)lk(f(λ))2=2∑λ∈Γ3,3(K7)lk(f(λ))2.
###### Proof of Theorem 2-3.

For vertices and of , we call the vertices the black vertices, the vertices the white vertices and the vertex the square vertex. Note that a -cycle of contains the square vertex if is odd. There are exactly seventy subgraphs of isomorphic to , because there are seven ways to choose the square vertex and ways to choose the remaining black and white vertices. Then for a spatial embedding of , by applying Theorem 2-2 (2) to the embedding restricted on , it follows that

 2∑γ∈Γ7(Gi)a2(f(γ))−4∑γ∈Γ6(Hi)a2(f(γ))−2∑γ∈Γ5(Gi)a2(f(γ)) = ∑λ∈Γ3,4(Gi)lk(f(λ))2−1,

where is the subgraph of isomorphic to not containing the square vertex . Let us take the sum of both sides of (2) for all . Since each -cycle of is common for exactly seven ’s (there are seven ways to choose the square vertex from the vertices of and then the assignment of the black and white vertices is uniquely determined), we have

 (2.3)

Since for each -cycle of there exists the unique such that contains (the assignment of the black and white vertices is uniquely determined), we have

 (2.4) 70∑i=1∑γ∈Γ6(Hi)a2(f(γ))=∑γ∈Γ6(K7)a2(f(γ)).

Since each -cycle of is common for exactly ten ’s (there are five ways to choose the square vertex from the vertices of and two ways to choose the remaining black and white vertices), we have

 (2.5) 70∑i=1∑γ∈Γ5(Gi)a2(f(γ))=10∑γ∈Γ5(K7)a2(f(γ)).

Since each pair of two disjoint cycles in is common for exactly six ’s (there are three ways to choose the square vertex from the 3-cycle in and two ways to choose the remaining black and white vertices), we have

 (2.6) 70∑i=1∑λ∈Γ3,4(Gi)lk(f(λ))2=6∑λ∈Γ3,4(K7)lk(f(λ))2.

Thus by combining (2.3), (2.4), (2.5) and (2.6) with (2), we have

 7∑γ∈Γ7(K7)a2(f(γ))−2∑γ∈Γ6(K7)a2(f(γ))−10∑γ∈Γ5(K7)a2(f(γ)) = 3∑λ∈Γ3,4(K7)lk(f(λ))2−35.

Then by (2) and Theorem 2-2 (1), it follows that

 4∑γ∈Γ6(K7)a2(f(γ))−8∑γ∈Γ5(K7)a2(f(γ)) = ∑λ∈Γ3,4(K7)lk(f(λ))2−14.

On the other hand, by Lemma 2-1 (1) we have

 2∑γ∈Γ6(K7)a2(f(γ))−4∑γ∈Γ5(K7)a2(f(γ)) = ∑λ∈Γ3,3(K7)lk(f(λ))2−7.

By (2) and (2), we have the desired conclusion. ∎

###### Proof of Lemma 2-1 (2).

Note that each pair of two disjoint -cycles of is common for exactly subgraphs isomorphic to if . Then by applying Theorem 2-3 to the embedding restricted on each of the subgraphs of isomorphic to and taking the sum of both sides of (2.1) over all of them, we have the result. ∎

Now we show the lemma which plays a major role in the proof of Theorem 1-3. The proof is in the same spirit as the proof of Theorem 2-2 (1) in [24].

###### Lemma 2-4.

Let be an integer. Assume that there exist three constants and such that

 ∑γ∈Γn−1(Kn−1)a2(g(γ))+b∑γ∈Γ5(Kn−1)a2(g(γ))=c∑λ∈Γ3,3(Kn−1)lk(g(λ))2+d

for any spatial embedding of . Then it follows that

 ∑γ∈Γn(Kn)a2(f(γ))+b(n−5)∑γ∈Γ5(Kn)a2(f(γ)) = c(n−6)(n+1)−3bn∑λ∈Γ3,3(Kn)lk(f(λ))2+d(n−1)+b2(n−15)

for any spatial embedding of .

###### Proof.

In the following, we denote the edge of connecting two vertices and by , and denote a path of length of consisting of two edges and by . We denote the subgraph of obtained from by deleting the vertex and all of the edges incident to by . Actually is isomorphic to for any . For and , let be the subgraph of obtained from by deleting the edges and . Note that is homeomorphic to , namely is obtained from by subdividing the edge by the vertex , see Fig. 2.2.

Let be a spatial embedding of . Then for the embedding restricted on , by the assumption it follows that

 ∑γ∈Γn(F(m)ij)a2(f(γ))+∑γ∈Γn−1(K(m)n−1)¯¯¯¯ij⊄γa2(f(γ)) = c{∑λ=γ∪γ′∈Γ3,4(F(m)ij)γ∈Γ4(F(m)ij), γ′∈Γ3(F(m)ij)¯¯¯¯¯¯¯¯imj⊂γlk(f(λ))2+∑λ∈Γ3,3(K(m)n−1)¯¯¯¯ij⊄λlk(f(λ))2}+d.

Let us take the sum of both sides of (2) over and . For a -cycle of , let and be the two vertices of which are adjacent to in ( and ). Then is a -cycle of . This implies that

 (2.11) ∑1≤i

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

 ∑1≤i

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

 (2.13) ∑1≤i

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

 (2.14) ∑1≤i

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

 ∑1≤i

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

 ∑1≤i