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 (2 × 1)-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 7 To whom questions concerning the numerical studies should be addressed. 8 To whom questions concerning the analytical studies should be addressed.

Characterising knotting properties of polymers in nanochannels

“…For example, prime knots in the 2 × 1 tube are 2-bridge knots. By applying the method used in [8], we obtain the following results about which knot patterns can occur in T L,M (see the appendix for some details of the proofs). Note that this result, while definitive for the case of non-local knot patterns, leaves open the possibility that for a given knot K which is embeddable in T L,M there might not be an associated local knot pattern in T L,M .…”

“…Figure 2(c) shows a local knot pattern in the same tube; note that the span of this local knot pattern is one greater than that shown in Figure 2(b). For the 2 × 1 tube, the arguments used in [8] can also be extended to prove that this difference in span holds for the smallest knot patterns of knots with up to 5 crossings. In summary the following result can be proved.…”

“…The trunk of a knot or link K is an invariant defined by trunk(K) = min E max t∈R |h −1 (t) ∩ E|, where E is any embedding of K in R 3 and h : R 3 → R is any given height function [15,8]. For another invariant, the bridge number b(K) of K, trunk(K) satisfies trunk(K) ≤ 2b(K).…”

“…By [8,Theorem 1], we can construct a polygon of knot type K in T L,M if and only if trunk(K) < (L + 1)(M + 1). Then we can obtain a proper knot pattern from such a polygon by opening its ends, i.e.…”

Knotting statistics for polygons in lattice tubes

“…This cannot however be true when restricted to finite-size tubes, since for any given L, M there will be infinitely many prime knot types K which cannot be embedded in the L × M tube, so C K = 0. (Indeed in [47] it has been established that only knots whose trunk is less than (L + 1)(M + 1) are embeddable in an L × M tube.) This does however also suggest that even in tubes where knots of type K can be embedded, we should not expect amplitude ratios to be universal, either between different tube sizes or between tubes and the full lattice.…”