A diagrammatic Alexander invariant of tangles

TL;DR

This paper introduces a novel diagrammatic approach to the Alexander polynomial, extending it from knots and links to a vector-valued invariant for tangles, enhancing its applicability in knot theory.

Contribution

It presents a new diagrammatic construction of the Alexander invariant and generalizes it to a vector-valued form for tangles, expanding the invariant's scope.

Findings

Provides a new diagrammatic construction of the Alexander polynomial.

Extends the invariant to a vector-valued form for tangles.

Enhances the tools available for studying knot and tangle invariants.

Abstract

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.