2020/07/06 by Yuanan Diao, Diao, Yuanan, Michael Finney +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57M25 #math.CO #math.GT #msc:57M25
paper · pdf · doi:10.48550/arxiv.2007.02819
16 pages, 8 figures, 1 table
arxiv created 2020/07/06 · arxiv updated 2020/07/07
Let Un be the set of un-oriented and rational links with crossing number n, a precise formula for |Un| was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let Λn be the set of oriented rational links with crossing number n and let Λn(d) be the set of oriented rational links with crossing number n (n≥ 2) and deficiency d. In this paper, we derive precise formulas for |Λn| and |Λn(d)| for any given n and d and show that Λn(d)=Fn-d-1(d)+\frac1+(-1)nd2F(\lfloor (d)/(2)\rfloor)\lfloor (n)/(2)\rfloor -\lfloor (d+1)/(2)\rfloor, where Fn(d) is the convolved Fibonacci sequence.