Arithmetic Kei Theory

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 ${\text{col}}_{\mathcal{K}}(n)$ of square-free integers. This is achieved by defining a fundamental kei for every such $n$. We conjecture that the asymptotic average order of ${\text{col}}_{\mathcal{K}}$ 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.