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 $$ \Lambda_n(d)=F_{n-d-1}^{(d)}+\frac{1+(-1)^{nd}}{2}F^{(\lfloor \frac{d}{2}\rfloor)}_{\lfloor \frac{n}{2}\rfloor -\lfloor \frac{d+1}{2}\rfloor}, $$ where $F_n^{(d)}$ is the convolved Fibonacci sequence.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2020
- DOI:
- 10.48550/arXiv.2007.02819
- arXiv:
- arXiv:2007.02819
- Bibcode:
- 2020arXiv200702819D
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Combinatorics;
- Primary 57M25
- E-Print:
- 16 pages, 8 figures, 1 table