Properties for a Certain Family of Knot Diagrams

In this note, we consider arithmetic properties of the function K(n) = (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: • K(n) is never odd, • K(n) is never a quadratic residue modulo 3, and • K(n) is never a quadratic residue modulo 5.