The number of oriented rational links with a given deficiency number

Abstract

Let $U_n$ be the set of un-oriented and rational links with crossing number $n$, a precise formula for $|U_n|$ was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let $Λ_n$ be the set of oriented rational links with crossing number $n$ and let $Λ_n(d)$ be the set of oriented rational links with crossing number $n$ ($n\ge 2$) and deficiency $d$. In this paper, we derive precise formulas for $|Λ_n|$ and $|Λ_n(d)|$ for any given $n$ and $d$ and show that $$ Λ_n(d)=F_{n-d-1}^{(d)}+\frac{1+(-1)^{nd}}{2}F^{(\lfloor \frac{d}{2}\rfloor)}_{\lfloor \frac{n}{2}\rfloor -\lfloor \frac{d+1}{2}\rfloor}, $$ where $F_n^{(d)}$ is the convolved Fibonacci sequence.