Knot fertility and lineage

Abstract

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot H is a descendant of another knot K if H can be obtained from a minimal crossing diagram of K by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration.

The knots that lie above all shadows

“…Related work. Besides proving the result mentioned above on the shadows in Figure 1, Cantarella, Henrich, Magness, O'Keefe, Perez, Rawdon, and Zimmer investigate in [1] several problems related to Question 1.1, including an exhaustive analysis on shadows of minimal crossing diagrams of knots with crossing number at most 10.…”

“…Moreover, we only consider two particular types of tangles, illustrated in Figure 6. A tangle is of Type I if it consists of a string λ with endpoints (−1, −1, 0) and (−1, 1, 3), and a string ρ with endpoints (1, −1, 0) and (1,1,3), and it is of Type II if it consists of a string λ with endpoints (−1, −1, 0) and (−1, 1, 3), and a string ρ with endpoints (1, 1, 0) and (1, −1, 3).…”

“…The shadow U of a tangle T is its projection onto the xy-plane, without over/under information at the crossings. Thus, regardless of whether T is of Type I or II, U consists of an arc L (the projection of λ) with endpoints (−1, −1) and (−1, 1) and an arc R (the projection of ρ) with endpoints (1, −1) and (1,1). We refer the reader to Figure 7(a) (the part contained in ∆) for an illustration of a tangle shadow.…”

Performance of the Uniform Closure Method for open knotting as a Bayes-type classifier

“…Some of this effect is doubtless due to the presence of composite knot types in our data such as +3 1 #+3 1 . However, we suspect that other knots in the same family, such as +5 1 , also contain subarcs classified as the trefoil knot [7,9,17,23,30,33,35].…”

Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial

“…Originally envisioned as a way to distinguish different atomic properties [48], modern work has suggested that a classification system could assist in the understanding of glueball particles [20]. Since then a series of systems have been suggested for ordering or relating knots within these ever expanding tabulations [10,13].…”