$\triangle Y$-exchanges and the Conway-Gordon theorems
Ryo Nikkuni, Kouki Taniyama
TL;DR
This paper extends Conway-Gordon theorems to graphs derived from complete graphs on 6 or 7 vertices through $ riangle Y$-exchanges, revealing invariants related to linking numbers and Arf invariants.
Contribution
It introduces a Conway-Gordon type theorem applicable to graphs obtained via $ riangle Y$-exchanges from complete graphs on 6 or 7 vertices.
Findings
The sum of linking numbers modulo 2 is invariant under $ riangle Y$-exchanges.
The sum of Arf invariants modulo 2 remains consistent after $ riangle Y$-exchanges.
The results generalize Conway-Gordon theorems to a broader class of graphs.
Abstract
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 -exchanges.
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