Entanglement complexity of graphs in Z³

Abstract

In this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z3. What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type?We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length (n) of the embedding goes to infinity, at least linearly in n. One example of a good measure of knot complexity is crossing number.Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.

Topological Information Embodied in Local Juxtaposition Geometry Provides a Statistical Mechanical Basis for Unknotting by Type-2 DNA Topoisomerases

“…These efforts have made important advances. [50][51][52][53][54][55][56] Here, we consider single-loop (one ring polymer) conformations configured on simple cubic lattices. Each conformation consists of n beads, and a set of n bonds joining the beads together to form a closed circuit, which can be knotted or unknotted.…”

“…In this respect, our methodology is distinct, yet complementary to other coarse-grained modeling investigations of polymer entanglement. [50][51][52][53][54][55][56][62][63][64][65][66][67][68][69][70][71] Here, for a range of small loop sizes, Table 1 gives the exact numbers of conformations that resulted from each of these five preformed juxtapositions. Each count in the table corresponds to the number of one-loop conformations with at least one instance of the given juxtaposition.…”

On the dominance of trivial knots among SAPs on a cubic lattice

“…We expect that the model should be suitable for general and mathematical study. In fact, several rigorous results on knotting probability have been derived for the SAP on the cubic lattice [1,2,3,4]. Thus, the main motivation of the present research is to characterize the knotting probability of the SAP on the cubic lattice through numerical simulations.…”

The Lace Expansion and its Applications

“…Thus the connective constant µ also serves as the growth constant for star polymers. For d = 3, a closely related result is proved in [194]; the proof extends to general d ≥ 2 [193]. Also, (6.25) gives c…”