Arithmetic properties for a certain family of knot diagrams

Darrin D. Frey, James Allen Sellers

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)65-73
Number of pages9
JournalArs Combinatoria
Volume77
StatePublished - Oct 1 2005

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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