The growth of the number of prime knots

Abstract

A fundamental and interesting question in knot theory is:Question 1. How many prime knots of n crossings are there ?Over time, knot theorists have answered this question for n ≤ 13 by the method of exhaustion: one writes down a list of all possible knots of n crossings, and then works hard to eliminate duplications from the list [12]. A perhaps easier question is the following:

Models of random knots

“…It is known that there are exponentially many knots with n crossings (Ernst and Sumners 1987;Welsh 1991;Carl Sundberg and Morwen Thistlethwaite 1998), but the exact count is known only for small n (Jim Hoste et al 1998). The difficulties in recognition and enumeration of n-crossing knots make this model less suitable for precise computations, though it is known that most knots are not rational (Ernst and Sumners 1987), nor are most links alternating (Thistlethwaite 1998).…”

“…The difficulties in recognition and enumeration of n-crossing knots make this model less suitable for precise computations, though it is known that most knots are not rational (Ernst and Sumners 1987), nor are most links alternating (Thistlethwaite 1998).…”

“…See Ernst and Sumners (1987) and Diao et al (2010) for corresponding results. Star diagrams are obtained from ð2n þ 1Þ-petal diagrams by straightening the segments between petal tips.…”

Predicting Knot or Catenane Type of Site-Specific Recombination Products

“…9 For small values of the MCN there are not many knots or catenanes with a given value. However, the number of knots and catenanes with MCN = n grows exponentially as a function of n, 74 and there are 1,701,936 knots with MCN ≤ 16. 10 So knowing the MCN is not sufficient to determine the knot or catenane.…”

Knots of genus one or on the number of alternating knots of given genus

“…(We will in the following observe that the results of §3 suggest a similar(ly bad) situation for any fixed genus.) That there are exponentially many genus one knots for bounded crossing number follows from the fact that the Whitehead doubles of distinct knots are distinct, their crossing number is linearly bounded in the crossing number of their companion and that the number of knots of given crossing number has an exponential lower bound [11]. (On the other hand, there are at most exponentially many knots of fixed crossing number at all [35].)…”