A relationship between two graphical models of the Kauffman polynomial

TL;DR

This paper explores the relationship between two different graphical models of the Kauffman polynomial, revealing a one-to-many correspondence and a bijection with related models, enhancing understanding of their structural connections.

Contribution

It establishes a detailed relationship between two graphical models of the Kauffman polynomial and links trivalent and 4-valent models, clarifying their structural correspondence.

Findings

One-to-many correspondence between HJ and WF models.

Bijection between trivalent and 4-valent models.

Insights into structural relationships among graphical models.

Abstract

There are two oriented 4-valent graphical models for the Kauffman polynomial: one (HJ) is obtained by combining Jaeger's formula and Kauffman-Vogel model for the Homflypt polynomial; the other (WF) is obtained by combining Kauffman-Vogel model for the Kauffman polynomial and Wu's formula. The main goal of this paper is to explore the relationship between the two models. We find that there is an one-to-many correspondence between the terms of HJ model and the terms of WF model. In addition, we investigate the relation between trivalent graphical models and 4-valent graphical models of both the Homflypt and Kauffman polynomials, and observe that there is a bijection between the terms of the two models.