Minimal Knotting Numbers

Abstract

This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.

Minimal knotted polygons in cubic lattices

“…where K#L is the connected sum of the knot types K and L. Similar results are known in the FCC lattice: one has that p n (0 1 ) = 0 if n < 3, and p 3 (0 1 ) = 8. Similarly, p n (3 1 ) = 0 if n < 15 [37], while p 15 (3 1 ) = 64. Observe that in the FCC lattice, p n (K) is a function on N; p n (K): N → N. That is, there are polygons of odd length.…”

The Stick Number for the Simple Hexagonal Lattice

“…(1) For each arrangement of sticks on the boundary w-levels from Tables 1 and 2, construct the stars of O, T and B. The computer algorithm employed to check for knottedness used the procedure described in [3]. The program revealed that every representative was the unknot, which concludes the proof.…”

The compressibility of minimal lattice knots

“…In three dimensions a ring polymer may be knotted, and it is known that the topological properties of ring polymers have an important effect on entropy [8,5]. Lattice knots are now a standard model for the polymer entropy problem in knotted ring polymers [22,26,10] and these objects have been the subject of numerous studies over the last two decades [19,12,27,21,11,18,25,23]. One of the advantages in models of lattice knots is that they are effective numerical models of simulating the effects of topological constraints (knotting) on the properties of ring polymers, and although one may not obtain direct quantitative results for real ring polymers, qualitative results may be examined to gain insight into the physical properties of ring polymers generally.…”

“…Amongst knot types of 8 crossing the knot types 8 18 In most cases the maximal compressibility is at zero pressure, but there are some exceptions to this in tables 9 and 10. For example, the maximum compressibility of the unknot is at p = 0.08664 .…”