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 ${\Lambda }_n$ be the set of oriented rational links with crossing number $n$ and let ${\Lambda }_n(d)$ 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 $|{\Lambda }_n|$ and $|{\Lambda }_n(d)|$ for any given $n$ and $d$ and show that $${\Lambda }_n(d)=F_{n-d-1}^{(d)}+\frac{1+(-1{)}^{nd}}{2}{F}_{\lfloor \frac{n}{2}\rfloor -\lfloor \frac{d+1}{2}\rfloor }^{(\lfloor \frac{d}{2}\rfloor )},$$ where $F_n^{(d)}$ is the convolved Fibonacci sequence.

16 pages, 8 figures, 1 table