2012/12/11 by Marc Soret, Soret, Marc, Marina Ville +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Connective tissue disorders research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1212.2347
openalex publication_date 2012/12/11 · openalex created_date 2016/09/23 · openalex updated_date 2026/07/28
A minimal knot is the intersection of a topologically embedded branched minimal disk in ℝ4 ℂ2 with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the embedding of the disk in ℂ2 are enough to determine the knot type, we talk of a simple minimal knot. Such a knot is given by three integers N < p,q; denoted by K(N,p,q), it can be parametrized in the cylinder as eiθ↦ (eNiθ,sin qθ,cos pθ). From this expression stems a natural representation of K(N,p,q) as an N-braid. In this paper, we give a formula for its writhe number, i.e. the signed number of crossing points of this braid and derive topological consequences. We also show that if q and p are not mutually prime, K(N,p,q) is periodic. Simple minimal knots are a generalization of torus knots.