A new method for the recognition and size characterization of a knot in a ring molecule placed upon a lattice

We show how a knot in a ring located upon a lattice may be characterized by two different measures of the perimeter length and by the mean square radius of gyration. This characterization is performed after the knot in a well-extended form has been reduced in size by a process that involves bead elimination and bead movement, movements being governed alternately by two criteria, the need to reduce the total length of the perimeter (or at least not to increase it) and the need to move a bead closer to the center of gravity of the object, so that elimination may subsequently take place. During contractions the beads must occupy different sites upon the lattice and bonds between pairs of beads are not permitted to pass through each other, so their topology is maintained. For the first few knots of the knot table the number of bonds, Nb (or the number of beads) is useful in discriminating, though unique values are found only for the 3(1) and 4(1) knots. The mean number of bonds within a contracted knot is a good measure of the crossing number, as also are the second and third measures of knot size, the mean perimeter length, and the radius of gyration. None of these measures are capable of readily distinguishing between different knots with the same number of crossings. Compacted knots themselves have a number of configurations upon the lattice that have interesting properties. (C) 1996 American Institute of Physics.