The Knot Entropy Conjecture states that the exponential growth rate of the number of n -edge lattice polygons with knot-type K is the same as that for unknot polygons. Moreover, the next order growth follows a power law in n with an exponent that increases by one for each prime knot in the knot decomposition of K . We provide the first proof of this conjecture by considering knots and non-split links in tube T ∗ , an ∞ × 2 × 1 sublattice of the simple cubic lattice. We establish upper and lower bounds relating the asymptotics of the number of n -edge polygons with fixed link-type in T ∗ to that of the number of n -edge unknots. For the upper bound, we prove that polygons can be unknotted by braid insertions. For the lower bound, we prove a pattern theorem for unknots using information from exact transfer-matrices. This work provides new knot theory results for 4-plats and new combinatorics results for lattice polygons. Connections to modelling polymers such as DNA in nanochannels are highlighted.

Hierarchical caching enables users to obtain content from one of (possibly) many caches between an edge router cache and an origin server, reducing latency/overall network traffic. This can be used in many network architectures, including P2P networks and Content Distribution Networks. Of particular interest are Information Centric Networks (ICNs), which decouple content identifiers from specific network hosts and explicitly consider universal caching (collaboration between network routers) as a desirable feature.Performance analysis of a large-scale hierarchy of caches requires accurate mathematical models for various cache replacement algorithms. There is no previous study that models LRU-k. This cache replacement algorithm is important since its principle is the basis of recent algorithms such as k-LRU that outperform LRU in many situations. We first model LRU-2 using Che's approximation, as a specific case of LRU-k for k=2. We also extend our model to a hierarchical network of LRU-2 caches. The model is validated analytically and with simulation. The experiments show that the proposed model approximates LRU-2 accurately. LRU-2 and 2-LRU are also compared analytically and with simulations. The comparison between the two algorithms illustrates that 2-LRU outperforms LRU-2.

We study the behaviour of prudent, perimeter and quasi-prudent self-avoiding walks and polygons in both two and three dimensions, as well as some solvable subsets. Our analysis combines exact solutions of some simpler cases, careful asymptotic analys