Bounds for minimum step number of knots confined to tubes in the simple cubic lattice

Abstract

Knots are ubiquitous in nature and their analysis has important implications in a wide variety of fields including fluid dynamics, material science and molecular and structural biology. In many systems particles are found in crowded environments hence it is natural to rigorously characterize the properties of knots in confined volumes. In this work we combine analytical and numerical work on the simple cubic lattice to determine the minimal number of lattice steps, minimum step number, needed to make a knot inside a tubular region. Our complementary approaches help us establish a detailed enumeration of minimal knot lengths and/or conformations of knots in tubular regions. Analytical results characterize the types of knots and links that can be embedded in a tubular regions and determines the minimum number of steps required to construct all 2-bridge knots and links up to ten crossings in the -tube. Simulation results, on the other hand, estimate the minimum step number and provide exact trajectories of all knot types up to eight crossings for wider tubular regions. These findings not only determine what knots and links can be built in a highly confined volume but also provide further evidence that the minimum step number required to realize a knot type increases with confining volume.