Knot projections with a single multi-crossing

Abstract

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 ubercrossing 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 ofübercrossing 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: theübercrossing numberü(K), and petal number p(K). These are the least number of loops in anyübercrossing or petal projection of K, respectively. We relateü(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.

Models of random knots

“…It is of interest to understand the relation between the crossing number c(K) of a knot and the least number of petals p(K) needed to represent it. We show in EvenZohar et al (2017a) that pðKÞ OðcðKÞÞ, and this bound is tight by results of Adams et al (2015a). They have also shown that cðKÞ Oðp 2 ðKÞÞ, which is also tight.…”

“…A random knot in the star model is generated by randomizing the ðn À 1Þð2n þ 1Þ crossings. Star diagrams are plane isotopic to closed n-braids (Adams et al 2015a), as demonstrated in Fig. 12b, c. The star model yields all knots, since the Petaluma model does, but with quite different distribution.…”

“…We start with the Petaluma model, studied by the author and collaborators (Even-Zohar 2017; , 2017a. Adams et al (2015a, b) have shown that every knot or link can be positioned so that its planar projection is injective except for a single point. Several projected strands may smoothly traverse this point of the über-crossing projection, each originating at a different height.…”

“…5. As observed in Adams et al (2015a), the Petaluma model is contained in the grid model, and obtained by conditioning on qðkÞ ¼ nk mod ð2n þ 1Þ.…”

Invariants of Random Knots and Links

“…Common models are based on random 4-valent planar graphs with randomly assigned crossings, random diagrams on the integer grid in R 2 , Gaussian random polygons [10,5,29], and random walks on lattices in R 3 [35,33]. While many interesting numerical studies have been performed, and interesting results obtained in these models, there have been few rigorous derivations of associated statistical measures.In this paper we study a model of random knots and links called the Petaluma model, based on the representation of knots and links as petal diagrams that was introduced by Adams and studied in [2]. The Petaluma model has the advantage of being both universal, in that it represents all knots and links, and combinatorially simple, so that knots have simple descriptions in terms of a single permutation.…”