Yamada Polynomial and associated link of $\theta$-curves

TL;DR

This paper explores the relationship between the Yamada polynomial of $ heta$-curves and the Jones polynomial of associated links, revealing their equivalence in brunnian cases and enhancing invariants for spatial graph analysis.

Contribution

It establishes a connection between the Yamada polynomial and the Jones polynomial for $ heta$-curves, especially showing their equivalence in brunnian cases, and examines the Jaeger polynomial as a related invariant.

Findings

Yamada polynomial and Jones polynomial are equivalent for brunnian $ heta$-curves

Associated links provide a useful invariant for $ heta$-curves

The Jaeger polynomial relates to the Yamada polynomial as a specialization.

Abstract

The discovery of polynomial invariants of knots and links, ignited by V. F. R. Jones, leads to the formulation of polynomial invariants of spatial graphs. The Yamada polynomial, one of such invariants, is frequently utilized for practical distinguishment of spatial graphs. Especially for $\theta$-curves, the polynomial is an ambient isotopy invariant after a normalization. On the other hand, to each $\theta$-curve, a 3-component link can be associated as an ambient isotopy invariant. The benefit of associated links is that invariants of links can be utilized as invariants of $\theta$-curves. In this paper we investigate the relation between the normalized Yamada polynomial of $\theta$-curves and the Jones polynomial of their associated links, and show that the two polynomials are equivalent for brunnian $\theta$-curves as a corollary. For our purpose the Jaeger polynomial of spatial graphs is observed, a specialization of which is equivalent to the Yamada polynomial.