Random Walk Invariants of String Links from R-Matrices

TL;DR

This paper explores the algebraic properties of string link invariants derived from R-matrices, revealing their structure as graded components of a tangle functor related to the Alexander Polynomial, and discusses their implications.

Contribution

It establishes an isomorphism between the exterior powers of the random walk invariant and graded components of the Alexander Polynomial-based tangle functor, providing new insights into string link invariants.

Findings

Exterior powers of the invariant are isomorphic to graded components of the tangle functor.

Properties of these representations of string link monoids are analyzed.

The invariants relate to the Alexander Polynomial through a specific division by the zero graded invariant.

Abstract

We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed.