Closed-Form Associators in Permutative Chord Diagrams

TL;DR

This paper derives a simplified, explicit closed-form solution to key equations in a restricted algebra of chord diagrams, aiding the understanding of knot invariants and properties.

Contribution

It provides a shorter, alternative derivation of Kurlin's equation by reducing complex algebraic equations to a simpler form in a permutative subalgebra.

Findings

Reduction of hexagon and pentagon equations to a simpler power series equation

Equivalence of reduction assumptions with Kurlin's approach

Explicit closed-form solution in a restricted algebraic setting

Abstract

Construction of a universal finite-type invariant can be reduced, under suitable assumptions, to the solution of certain equations (the hexagon and pentagon equations) in a particular graded associative algebra of chord diagrams. An explicit, closed-form solution to these equations may, indirectly, give information about various interesting properties of knots, such as which knots are ribbon. However, while closed-form solutions (as opposed to solutions which can only be approximated to successively higher degrees) are needed for this purpose, such solutions have proven elusive, partly as a result of the non-commutative nature of the algebra. To make the problem more tractable, we restrict our attention to solutions of the equations in the subalgebra of horizontal chord diagrams, viewed as a graded unital permutative algebra -- where `permutative' means that u[x,y]=0 whenever u has degree at least 1. We show that this restriction leads in a straightforward and fairly short way to a reduction of the hexagon and pentagon equations to a simpler equation taken over the algebra of power series in two commuting variables. This equation had been found and solved explicitly by Kurlin under a superficially different set of reduction assumptions, which we show here are in fact equivalent to ours. This paper thus provides an alternative (simpler and shorter) derivation of Kurlin's equation.