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 \(5\).