2009/07/31 by Ryo Nikkuni · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics #Discrete mathematics #Geometric and Algebraic Topology #Geometry #Graph #Homotopy and Cohomology in Algebraic Topology #Mathematical proof #Mathematics #Modulo #math.GT #msc:57M15 #msc:57M25
paper · pdf · doi:10.1016/j.topol.2009.08.013
published as Topology Appl. 156 (2009), 2782--2794 · 17 pages, 10 figures
arxiv created 2009/08/12 · openalex publication_date 2009/08/27 · arxiv updated 2020/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 Alfonsin and Huh-Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods.