Lattice Knots in a Slab

TL;DR

This study investigates minimal length lattice knots confined in slabs, verifying previous results, and models the entropic forces and work involved in squeezing knotted polymers between hard walls or by external forces.

Contribution

It provides new data on minimal lattice knot lengths, improves upon previous models, and analyzes the entropic forces and work during confinement of knotted polymers.

Findings

Minimal length data mostly confirm previous results, with one exception.

Models of grafted knotted polymers reveal force profiles and compressibility.

Total work done during squeezing to minimal state is quantified.

Abstract

In this paper the number and lengths of minimal length lattice knots confined to slabs of width $L$, is determined. Our data on minimal length verify the results by Sharein et.al. (2011) for the similar problem, expect in a single case, where an improvement is found. From our data we construct two models of grafted knotted ring polymers squeezed between hard walls, or by an external force. In each model, we determine the entropic forces arising when the lattice polygon is squeezed by externally applied forces. The profile of forces and compressibility of several knot types are presented and compared, and in addition, the total work done on the lattice knots when it is squeezed to a minimal state is determined.