2017/12/11 by Franck Ramaharo, Ramaharo, Franck, Fanja Rakotondrajao +1 · 2 citations
Computer Science · Mathematics · #52C30 #57M25 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #math.CO #msc:52C30 #msc:57M25 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1712.04026
24 pages, 18 figures
arxiv created 2017/12/11 · openalex publication_date 2017/12/11 · arxiv updated 2017/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We split the crossings of the foil knot and enumerate the resulting states with a generating polynomial. Unexpectedly, the number of such states which consist of two components are given by the lazy caterer's sequence. This sequence describes the maximum number of planar regions that is obtained with a given number of straight lines. We then establish a bijection between this partition of the plane and the concerned foil splits sequence.