A multi-variable Alexander polynomial for a framed transverse graph

TL;DR

This paper introduces a new multi-variable Alexander polynomial for framed, oriented transverse graphs in 3-sphere, extending classical invariants and proving its equivalence to a quantum group-based polynomial.

Contribution

It defines a rotation number for transverse graph diagrams and constructs a normalized polynomial invariant that generalizes classical invariants for graphs.

Findings

Invariant coincides with Viro's $U_q(rak{gl}(1|1))$-Alexander polynomial

Extends classical rotation number to transverse graphs

Provides a new tool for studying framed, oriented transverse graphs

Abstract

We propose a definition of the rotation number for transverse graph diagrams, extending the classical notion of the rotation number for plane curves. Using this, we introduce a normalized multi-variable Alexander polynomial for framed, oriented transverse graphs without sinks or sources, embedded in the 3-sphere $S^3$. We prove that our invariant coincides with the $U_q(\mathfrak{gl}(1\vert 1))$-Alexander polynomial proposed by Viro.