Entanglement complexity of spanning pairs of lattice polygons

TL;DR

The paper studies entanglement complexity of spanning pairs of lattice polygons (2SAPs) confined in lattice tubes, where the linking probability tends to 1 exponentially with size. It develops a general framework for good measures of spanning-link complexity based on $k$-tangle products and shows that, for tubes with $\min\{M,N\}\ge 1$ and $N+M\ge 4$, almost all large 2SAPs have complexity growing linearly with size. It introduces Equal-Height Trunk (EH-trunk) to constrain embeddability and proves that a nontrivial 2-component link embeds in an $M\times N$ tube with equal spans iff EH-trunk$(L) < (M+1)(N+1)$. The results connect classical knot invariants to good spanning-link measures and examine local versus non-local knot patterns, with implications for polymer confinement and nanopore translocation.

Abstract

We study the entanglement complexity of a system consisting of two simple-closed curves (self-avoiding polygons) that span a lattice tube, referred to as a 2SAP. 2SAPs are of interest as the first known model of confined ring polymers where the linking probability goes to 1 exponentially with the size of the system. Atapour et al proved this in 2010 by showing that all but exponentially few sufficiently large 2SAPs contain a pattern that guarantees the 2SAP is non-split, provided that the requisite pattern fits in the tube. This result was recently extended to all tubes sizes that admit non-trivial links. Here we develop and apply knot theory results to answer more general questions about the entanglement complexity of 2SAPs. We first extend the 1992 concept of a good measure of knot complexity to a good measure, $F$, of spanning-link complexity for $k$-component links. Using tangle products, we show, for example, that the more complex the prime knot decomposition of any component of a given link type, the greater its $F$-measure. We then prove that all but exponentially few size $m$ 2SAPs have $F$ complexity that grows at least linearly in $m$ as $m\to \infty$. We establish that good measures of knot complexity yield good measures of spanning-link complexity. We also establish conditions whereby more general link invariants can yield good measures. In particular, we establish that measures based on several classical invariants are good measures by our definition, eg bridge number or the number of $p$-colourings. Finally, we consider how the tube dimensions affect which links are embeddable as 2SAPs as well as geometric restrictions on the entanglement complexity of the embeddings. For example, we establish that there are two-component links that occur as 2SAPs in a given tube size only when one of the components is forced into a non-minimal bridge number conformation.

Figures (19)

• Figure 1: (a) An embedding of a 2-component unlink in a $(2\times 3)$ lattice tube that is a 2SAP. In this case, the span of the 2SAP is 6. (b) An embedding of a 2-component unlink in a $(2\times 1)$ lattice tube that is not a 2SAP.
• Figure 2: (a) Two 2-component links (top) are concatenated together by component-wise connected sum operations to create a new link (bottom). This concatenation operation is equivalent to a 2-strand tangle product operation (middle) as in Definition \ref{['def:product']}. This example also illustrates that concatenating two unlinks can yield a non-split link. (b) Half of the concatenation algorithm described in Definition \ref{['def:concat']} for concatenating $k$SAPs, $k=2$: dashed lines represent a 2SAP $\theta_1$ and solid lines indicate half the edges added to concatenate to any other 2SAP.
• Figure 3: (a) The link $L(K)$ is the split union of $k$ knots (shown in black for $k=2$) where one is knot-type $K$ (localized in the dashed blue circle) and the others are the unknot ($0_1$). $L(K)$ is also used to represent the $k$-tangle triple $(L(K),G_1,G_2)$ where $G_1$ and $G_2$ are the $k$-star graphs shown in green and red respectively for the case $k=2$ (see Definition \ref{['def:tangle']} in Section \ref{['sec:linkGoodMeasures']}) (b) A pattern that increases complexities of 2SAPs. Using our terminology, this is $L(3_1)$.
• Figure 4: (a) Two $3$-tangle-triples $(L(K^1),G_1^1,G_2^1)$ and $(L(K^2),G_1^2,G_2^2)$ where $K^i$ for $i=1,2$ is any knot. (b) A result of performing a 3-strand tangle product to get $(L(K^1\#K^2),G_1^1,G_2^2)$.
• Figure 5: (a) The number of prime knot factors may decrease under concatenation. The resulting link after concatenation is the prime link $L10a38$ from Thistlethwaite's table of prime links. (b) A closure of this pattern is a prime link $L7a2.$

Theorems & Definitions (42)

• Theorem 1
• Corollary 0
• Theorem 2
• Theorem 3
• Corollary 0
• Theorem 4
• Theorem 5
• Definition 1
• Definition 2
• Theorem 6: Sot98atapour2010linking