Generalizations of the ConwayGordon 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 twocomponent 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 ConwayGordon 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 ConwayGordon 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 triangletriangle Hopf links.
Key words and phrases:
Spatial graphs, ConwayGordon theorems1991 Mathematics Subject Classification:
Primary 57M15; Secondary 57M251. 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 wellknown as the ConwayGordon theorem.
Theorem 11.
(ConwayGordon [8])

For any spatial embedding of , it follows that
where denotes the linking number in .

For any spatial embedding of , it follows that
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 11 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 ConwayGordon 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, §26]). On the other hand, as far as the authors know, there have been little results about a generalization of the ConwayGordon type congruences for complete graphs on eight or more vertices. Our purposes in this paper are to generalize the ConwayGordon 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 ConwayGordon theorem for which was proven by the second author as follows.
Theorem 12.
(Nikkuni [24]) For any spatial embedding of , it follows that
Note that Theorem 11 (1) can be recovered by taking the modulo two reduction of (12), namely Theorem 12 is an integral lift of Theorem 11 (1). In [24], an integral lift of Theorem 11 (2) was also given (see Theorem 22 (1)). In this paper, we generalize Theorem 12 for complete graphs with arbitrary number of vertices greater than or equal to six as follows.
Theorem 13.
Let be an integer. For any spatial embedding of , it follows that
Note that any pair of two disjoint cycles of is not common for any two different subgraphs of isomorphic to . Then Theorem 11 (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) 
Corollary 14.
Let be an integer. For any spatial embedding of , it follows that
Remark 15.
EndoOtsuki 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 14 is sharp. In particular for any cycle of , is a trivial knot. Thus it follows that for an integer ,
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 13, we also have the following congruence, that is a generalization of Theorem 11 (2).
Corollary 16.
Let be an integer. For any spatial embedding of , it follows that
Proof of Corollary 16.
For any two spatial embeddings and of , by Theorem 13, we have
Since and have the same parity, that is equal to the parity of , by (1), it follows that
(1.5) 
Note that there exists a spatial embedding of such that
(1.6) 
see Remark 15 or Remark 112. Thus by (1.5) and (1.6), it follows that
(1.7) 
for any spatial embedding of . If , since is even, by (1.7), we have
If , since is even and is odd, by (1.7), we have
If , since is odd and is even, by (1.7), we have
This completes the proof. ∎
Remark 17.
Theorem 13 (and Corollary 14) 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], AdamsBrennanGreilsheimerWoo [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 18.
Let be a link. Then the followings hold.

If is a nontrivial knot, then .

if and only if is equivalent to , or .

if and only if is equivalent to or .

if and only if is equivalent to , , , , , a granny knot , a square knot , , or .
Proposition 18 (1) says that every polygonal knot with less than or equal to five sticks is trivial. Thus for rectilinear spatial complete graphs, by Theorem 13 we have the following immediately.
Theorem 19.
Let be an integer. For any rectilinear spatial embedding of , it follows that
As we saw in Proposition 18 (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 “triangletriangle” 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 110.
Theorem 110 implies that is less than or equal to , and by a direct calculation we have
(1.8) 
Thus by (1.3), (1.8) and Theorem 19, we have the following.
Corollary 111.
Let be an integer. For any rectilinear spatial embedding of , it follows that
Remark 112.
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 triangletriangle Hopf links. Thus the lower bound in Corollary 111 is sharp. But the authors strongly think that it cannot be expected at all that the upper bound is sharp if , see Example 32.
As an application of Corollary 111, 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 111 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 PolyakViro [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 113.
Let be a knot whose stick number is less than or equal to . Then it follows that
By Corollary 111 and Proposition 113, 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 114.
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 34 and Remark 35), 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 13. In section , we give examples and present some problems.
2. Proof of Theorem13
We show some lemmas which are needed to prove Theorem 13.
Lemma 21.

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

Let be an integer. For any spatial embedding of , it follows that
Proof of Lemma 21 (1).
In order to prove Lemma 21 (2), we recall integral ConwayGordon 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 22.
Then by applying Theorem 22 (2) to each of the subgraphs of isomorphic to and combining with Theorem 22 (1), we also have the following equation for every spatial embedding of .
Theorem 23.
For any spatial embedding of , it follows that
(2.1) 
Proof of Theorem 23.
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 22 (2) to the embedding restricted on , it follows that
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) 
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) 
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 3cycle in and two ways to choose the remaining black and white vertices), we have
(2.6) 
Thus by combining (2.3), (2.4), (2.5) and (2.6) with (2), we have
Then by (2) and Theorem 22 (1), it follows that
On the other hand, by Lemma 21 (1) we have
Proof of Lemma 21 (2).
Now we show the lemma which plays a major role in the proof of Theorem 13. The proof is in the same spirit as the proof of Theorem 22 (1) in [24].
Lemma 24.
Let be an integer. Assume that there exist three constants and such that
for any spatial embedding of . Then it follows that
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
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) 
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
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) 
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) 
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
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