2015/03/01 by Colin Adams, Thomas Crawford, Benjamin DeMeo +6 · 3 citations
Mathematics · Computer Science · Medicine · #Geometric and Algebraic Topology #semigroups and automata theory #Botulinum Toxin and Related Neurological Disorders #Mathematics #Knot (papermaking) #Knot invariant #Crossing number (knot theory) #Combinatorics #Tricolorability #Projection (relational algebra) #Skein relation #Knot theory #Algorithm
paper · doi:10.1142/s021821651550011x
openalex publication_date 2015/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
An n-crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an übercrossing projection, a knot projection with a single n-crossing. Such a projection is necessarily composed of a collection of loops emanating from the crossing. We prove the surprising fact that all knots have a special type of übercrossing projection, which we call a petal projection, in which no loops contain any others. The rigidity of this form allows all the information about the knot to be concentrated in a permutation corresponding to the levels at which the strands lie within the crossing. These ideas give rise to two new invariants for a knot K: the übercrossing number ü(K), and petal number p(K). These are the least number of loops in any übercrossing or petal projection of K, respectively. We relate ü(K) and p(K) to other knot invariants, and compute p(K) for several classes of knots, including all knots of nine or fewer crossings.