Enumerating the states of the twist knot

2017/12/18 by Franck Ramaharo, Ramaharo, Franck
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #57M25 #94B25 #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Orthopedic Surgery and Rehabilitation #dental development and anomalies #math.CO #msc:57M25 #msc:94B25

paper · pdf · doi:10.48550/arxiv.1712.06543

31 pages, 23 figures

arxiv created 2017/12/18 · openalex publication_date 2017/12/18 · arxiv updated 2017/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We enumerate the state diagrams of the twist knot shadow which consist of the disjoint union of two trivial knots. The result coincides with the maximal number of regions into which the plane is divided by a given number of circles. We then establish a bijection between the state enumeration and this particular partition of the plane by means of binary words.

Citations

• A state enumeration of the foil knot

Related

• A state enumeration of the foil knot
• Statistics on some classes of knot shadows
• The knots that lie above all shadows
• Encoding knots by clasp diagrams
• The twisting number of a ribbon knot is bounded below by its doubly slice genus