No abstract

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numerically, the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Our analytical work is based on improvement of a method introduced by Diao to enumerate conformations of a given knot type for a fixed length. This method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 (which is 24) to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34, respectively. Using an independent method based on the BFACF algorithm, we provide a complete list of numerical estimates (upper bounds) of the minimum step numbers for prime knots up to ten crossings, which are improvements over current published numerical results. We enumerate all minimum lattice knots of a given type and partition them into classes defined by BFACF type 0 moves.

No abstract

We give a brief introduction to the software KnotPlot. The goals of this chapter are twofold: 1) to help a new user get started with using KnotPlot and 2) to provide veteran users with additional background and functionality available in the software. This chapter is not linear. Each section is generally self contained. The list below provides a short description of what is in each section. We highlight three particular sections: Section 9 that lists and describes commonly used commands, Section 10 that shows the user how to change parameter values (like the background color), and Section 16 that lists commonly used commands in popular user activities. -Setting it up:Downloading and setting up KnotPlot 3 -Loading and saving: Basic input/output for knot/link models 4 -Changing the view or embedding: Rotating a model 5 -Relaxing knots and links examples: How to relax a knot/link so that it is visually appealing 6 -Making pictures: