An Alexander polynomial for MOY graphs

TL;DR

This paper introduces a new Alexander polynomial invariant for MOY graphs, refining previous constructions and establishing MOY-type relations that characterize the invariant and enable a graphical link polynomial definition.

Contribution

The authors define a novel Alexander polynomial for MOY graphs, establish MOY-type relations, and connect it to link invariants through a graphical approach.

Findings

The invariant satisfies MOY-type relations.

The relations uniquely determine the Alexander polynomial.

Applications include properties and potential uses in knot theory.

Abstract

We introduce an Alexander polynomial for MOY graphs. For a framed trivalent MOY graph $\mathbb{G}$, we refine the construction and obtain a framed ambient isotopy invariant $\Delta_{(\mathbb{G},c)}(t)$. The invariant $\Delta_{(\mathbb{G}, c)}(t)$ satisfies a series of relations, which we call MOY-type relations, and conversely these relations determine $\Delta_{(\mathbb{G}, c)}(t)$. Using them we provide a graphical definition of the Alexander polynomial of a link. Finally, we discuss some properties and applications of our invariants.