Hiroka Hashimoto, Ryo Nikkuni
For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of the Conway polynomial.
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We give a Conway-Gordon type formula for invariants of knots and links in a spatial complete four-partite graph $K_{3,3,1,1}$ in terms of the square of the linking number and the second coefficient of the Conway polynomial. As an application, we show that every rectilinear spatial $K_{3,3,1,1}$ contains a nontrivial Hamiltonian knot.
Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this paper, we give a Conway-Gordon type theorem for any graph which is obtained from the complete graph on 6 or 7 vertices by a finite sequence of $\triangle Y$-exchanges.
In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. 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-Ramirez Alfo…
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 that 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 vertice…
This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more explicit relationship for the graph $K_{3,3,1}$ between knotting and linking, which relates the sum of the squares of linking numbers of links in the embedding and the second coefficient of the Conway polynomial of particular cycles in the embedding.
A short proof of the generalized Conway--Gordon--Sachs theorem
本文针对广义Conway--Gordon--Sachs定理，在整数环上给出了一个比现有证明更简短的证明，该定理将六顶点完全图的情形推广到任意n个顶点的完全图。
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 Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determin…
In this paper we present new proofs of the Conway-Gordon-Sachs and Sachs Theorems on the linked cycles in graphs embedded in $\R^3$. We reduce these theorems to certain property of graphs mapped to the plane.
We exhibit relations between van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems, by presenting direct proofs of some implications between them. The key idea is an interesting relation between the van Kampen and the Conway-Gordon-Sachs numbers for restrictions of a map of $(d+2)$-simplex to $\mathbb R^d$ to the $(d+1)$-face and to the $[d/2]$-skeleton.
Sachs and Conway-Gordon used linking number and a beautiful counting argument to prove that every graph in the Petersen family is intrinsically linked (have a pair of disjoint cycles that form a nonsplit link in every spatial embedding) and thus each family member has no flat spatial embedding (an embedding for which every cycle bounds a disk with interior disjoint from the graph). We give an alternate proof that every Petersen family graph has no flat embedding by applying Böhme's Lemma and the Jordan-Brouwer Separation Theorem.
Converses to generalized Conway--Gordon type congruences
针对空间完全图的 Conway–Gordon 型同余，本文证明其逆命题：所有满足该同余的整数均可由某个空间完全图实现。
In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any $n+3$ general position points in $\mathbb{R}^n$. If $n$ is odd, then there are two linked $(n+1)/2$-simplices with the vertices at these points. If $n$ is even, then one can choose two disjoint $(n+2)/2$-tuples such that the interiors $(n/2)$-simplices with the vertices at these $(n+2)/2$-tuples intersect each other.} This theorem is interesting even in case of small dimensions.
In 1983 Conway and Gordon proved that any embedding of the complete graph $K_7$ into $\mathbb{R}^3$ contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper we are interested in knotted Hamiltonian cycles in linear embedding of $K_7$. Concretely it is shown that any linear embedding of $K_7$ contains at most three figure-8 knots as its Hamiltonian cycles.
Linking invariants of spatial graphs
该研究回顾空间图的环绕数与Wu–Simon数定义，揭示Conway–Gordon–Sachs定理的逆命题，提出其向多重环绕推广的猜想，论述基于平面图表，无需空间几何知识。
Some `converses' to intrinsic linking theorems
本文研究了内在链接定理的逆定理，证明了高维情况下图K6的链接系数特性及相关结论。
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