2014/11/07 by Kyungpyo Hong, Hyoung-Jun Kim, Hong, Kyungpyo +5 · 1 citation
Computer Science · Dentistry · Mathematics · #57M25 #57M27 #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Oral and Maxillofacial Pathology
paper · pdf · doi:10.48550/arxiv.1411.1845
openalex publication_date 2014/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Len(K) be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for Len(K) of a nontrivial knot K in terms of its crossing number c(K) as follows: Len(K) ≤ min \ (3)/(4)c(K)2 + 5c(K) + (17)/(4), (5)/(8)c(K)2 + (15)/(2)c(K) + (71)/(8) \. The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We also provide upper bounds for the minimum ropelength Rop(K) which is close to twice Len(K): Rop(K) ≤ min \ 1.5 c(K)2 + 9.15 c(K) + 6.79, 1.25 c(K)2 + 14.58 c(K) + 16.90 \.