The number of oriented rational links with a given deficiency number

TL;DR

This paper derives exact formulas for counting oriented rational links with a specified crossing number and deficiency, extending previous un-oriented link enumeration results.

Contribution

It provides the first precise enumeration formulas for oriented rational links with given crossing number and deficiency, involving convolved Fibonacci sequences.

Findings

Formulas for the number of oriented rational links with crossing number n.

Explicit enumeration of oriented rational links with a fixed deficiency d.

Use of convolved Fibonacci sequences in link enumeration.

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 $$ \Lambda_n(d)=F_{n-d-1}^{(d)}+\frac{1+(-1)^{nd}}{2}F^{(\lfloor \frac{d}{2}\rfloor)}_{\lfloor \frac{n}{2}\rfloor -\lfloor \frac{d+1}{2}\rfloor}, $$ where $F_n^{(d)}$ is the convolved Fibonacci sequence.