Arithmetic Properties for a Certain Family of Knot Diagrams

Abstract

In this note, we consider arithmetic properties of the function

$$K(n)=\frac{(2n)!(2n+2)!}{(n-1)!(n+1){!}^2(n+2)!}$$

which counts the number of two-legged knot diagrams with one self-intersection and $n-1$ tangencies. This function recently arose in a paper by Jacobsen and Zinn-Justin on the enumeration of knots via a transfer matrix approach. Using elementary number theoretic techniques, we prove various results concerning $K(n)$, including the following:

1. $K(n)$ is never odd,
2. $K(n)$ is never a quadratic residue modulo $3$, and
3. $K(n)$ is never a quadratic residue modulo $5$.