The generalized Yamada polynomials of virtual spatial graphs

TL;DR

This paper introduces a generalized Yamada polynomial for virtual spatial graphs, extending classical invariants to virtual knot theory and providing tools to detect non-classical virtual links.

Contribution

It defines a new polynomial invariant for virtual spatial graphs, proves its invariance under certain isotopies, and explores its relation to existing link polynomials.

Findings

The generalized Yamada polynomial is invariant under rigid and pliable vertex isotopies.

It can distinguish non-classical virtual links from classical ones.

A computational method using Mathematica is developed for calculating the polynomial.

Abstract

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is similar to the topological definition of a virtual link. Our main goal is to generalize the classical Yamada polynomial that is defined for a spatial graph. We define a generalized Yamada polynomial for a virtual spatial graph and prove that it can be normalized to a rigid vertex isotopic invariant and to a pliable vertex isotopic invariant for graphs with maximum degree at most 3. We consider the connection and difference between the generalized Yamada polynomial and the Dubrovnik polynomial of a classical link. The generalized Yamada polynomial specializes to a version of the Dubrovnik polynomial for virtual links such that it can be used to detect the non-classicality of some virtual links. We obtain a specialization for the generalized Yamada polynomial (via the Jones-Wenzl projector $P_2$ acting on a virtual spatial graph diagram), that can be used to write a program for calculating it based on Mathematica Code.