Arithmetic Kei Theory

TL;DR

This paper introduces a novel arithmetic kei theory linking algebraic structures called kei to number theory, defining new invariants for square-free integers and exploring their asymptotic behavior.

Contribution

It extends kei theory to number theory by defining coloring invariants for square-free integers and conjectures their average order relates to random braid colorings, with proofs for specific cases.

Findings

Defined a fundamental kei for square-free integers

Conjectured the asymptotic average order of coloring invariants

Proven the conjecture for several specific cases

Abstract

A kei, or 2-quandle, is an algebraic structure one can use to produce a numerical invariant of links, known as coloring invariants. Motivated by Mazur's analogy between prime numbers and knots, we define for every finite kei $\mathcal{K}$ an analogous coloring invariant $\textrm{col}_{\mathcal K}(n)$ of square-free integers. This is achieved by defining a fundamental kei for every such $n$. We conjecture that the asymptotic average order of $\textrm{col}_{\mathcal K}$ can be predicted to some extent by the colorings of random braid closures. This conjecture is fleshed out in general, building on previous work, and then proven for several cases.