On Yamada polynomial of spatial graphs obtained by edge replacements

TL;DR

This paper develops formulas for calculating the Yamada polynomial of complex spatial graphs created by replacing edges with spatial components, revealing dense zero distributions and linking to chain polynomials.

Contribution

It introduces new formulas for Yamada polynomials of spatial graphs formed by edge replacements and explores their zero distributions and relation to chain polynomials.

Findings

Zeros of Yamada polynomials are dense in a specific complex region.

Formulas relate Yamada polynomials to chain polynomials.

Spatial graph constructions influence polynomial properties.

Abstract

We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labelled graphs is obtained.