Meta-Monoids, Meta-Bicrossed Products, and the Alexander Polynomial

TL;DR

The paper introduces a new algebraic invariant for tangles that encompasses the classical Alexander polynomial and its multivariable extension, providing a computationally efficient and meaningful tool for knot and link analysis.

Contribution

It develops a novel algebraic framework called meta-monoids and meta-bicrossed products to understand and compute the Alexander polynomial and its extensions.

Findings

Contains classical Alexander polynomial as a special case

Provides a more meaningful and efficient computational approach

Extends to multivariable Alexander invariants for links

Abstract

We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of the computationally efficient members of the family of Alexander invariants, it is the most meaningful. These are lecture notes for talks given by the first author, written and completed by the second. The talks, with handouts and videos, are available at http://www.math.toronto.edu/drorbn/Talks/Regina-1206/. See also further comments at http://www.math.toronto.edu/drorbn/Talks/Caen-1206/#June8.