Knots cascade detected by a monotonically decreasing sequence of values.

Xin Liu,Renzo L Ricca

doi:10.1038/srep24118

Abstract

Due to reconnection or recombination of neighboring strands superfluid vortex knots and DNA plasmid torus knots and links are found to undergo an almost identical cascade process, that tend to reduce topological complexity by stepwise unlinking. Here, by using the HOMFLYPT polynomial recently introduced for fluid knots, we prove that under the assumption that topological complexity decreases by stepwise unlinking this cascade process follows a path detected by a unique, monotonically decreasing sequence of numerical values. This result holds true for any sequence of standardly embedded torus knots T(2, 2n + 1) and torus links T(2, 2n). By this result we demonstrate that the computation of this adapted HOMFLYPT polynomial provides a powerful tool to measure topological complexity of various physical systems.

Highlights

Our adapted HOMFLYPT polynomial we show that the particular cascade above can be detected by a unique, monotonically decreasing sequence of numerical values
Let us make the following assumptions: A1: all torus knots T(2, 2n + 1) and links T(2, 2n) considered here are standardly embedded on a mathematical torus in closed braid form; A2: all torus knots T(2, 2n + 1) and links T(2, 2n) form an ordered set {T(2, n)} (n ∈ ) of elements listed according to their decreasing value of topological complexity given by the minimum number of crossings cmin = n; A3: a ny topological transition between two contiguous elements of {T(2, n)} is determined by a single, anti-parallel reconnection event21
The results presented here hold true for any sub-sequence of standardly embedded torus knots and links in the family {T(2, n)}± under the assumption that any topological transition between contiguous knot/link types in {T(2, n)}± is produced by a single, anti-parallel reconnection event

Summary

Introduction

Our adapted HOMFLYPT polynomial we show that the particular cascade above can be detected by a unique, monotonically decreasing sequence of numerical values. Polynomial computations for knots and links are based on the recursive application of two skein relations, by the use of diagrams16 obtained from the indented projection of a given knot with minimal crossings (see top diagram of Fig. 3).

Results

Conclusion