Totally Twisted Khovanov Homology

We define a variation of Khovanov homology with an explicit description in terms of the spanning trees of a link projection. We prove that this new theory is a link invariant and describe some of its properties. Finally, we provide some the results of some computer computations of the invariant.Comment: 45 pages, 21 figure

Alternately-twisted cube as an interconnection network.

by Wong Yiu Chung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1991.Bibliography: leaves [100]-[101]AcknowledgementAbstractChapter 1. --- Introduction --- p.1-1Chapter 2. --- Alternately-Twisted Cube: Definition & Graph-Theoretic Properties --- p.2-1Chapter 2.1. --- Construction --- p.2-1Chapter 2.2. --- Topological Properties --- p.2-12Chapter 2.2.1. --- "Node Degree, Link Count & Diameter" --- p.2-12Chapter 2.2.2. --- Node Symmetry --- p.2-13Chapter 2.2.3. --- Sub cube Partitioning --- p.2-18Chapter 2.2.4. --- Distinct Paths --- p.2-23Chapter 2.2.5. --- Embedding other networks --- p.2-24Chapter 2.2.5.1. --- Rings --- p.2-25Chapter 2.2.5.2. --- Grids --- p.2-29Chapter 2.2.5.3. --- Binary Trees --- p.2-35Chapter 2.2.5.4. --- Hypercubes --- p.2-42Chapter 2.2.6. --- Summary of Comparison with the Hypercube --- p.2-44Chapter 3. --- Network Properties --- p.3-1Chapter 3.1. --- Routing Algorithms --- p.3-1Chapter 3.2. --- Message Transmission: Static Analysis --- p.3-5Chapter 3.3. --- Message Transmission: Dynamic Analysis --- p.3-13Chapter 3.4. --- Broadcasting --- p.3-17Chapter 4. --- Parallel Processing on the Alternately-Twisted Cube --- p.4-1Chapter 4.1. --- Ascend/Descend class algorithms --- p.4-1Chapter 4.2. --- Combining class algorithms --- p.4-7Chapter 4.3. --- Numerical algorithms --- p.4-8Chapter 5. --- "Summary, Comparison & Conclusion" --- p.5-1Chapter 5.1. --- Summary --- p.5-1Chapter 5.2. --- Comparison with other hypercube-like networks --- p.5-2Chapter 5.3. --- Conclusion --- p.5-7Chapter 5.4. --- Possible future research --- p.5-7Bibliograph

Interconnection Networks Embeddings and Efficient Parallel Computations.

To obtain a greater performance, many processors are allowed to cooperate to solve a single problem. These processors communicate via an interconnection network or a bus. The most essential function of the underlying interconnection network is the efficient interchanging of messages between processes in different processors. Parallel machines based on the hypercube topology have gained a great respect in parallel computation because of its many attractive properties. Many versions of the hypercube have been introduced by many researchers mainly to enhance communications. The twisted hypercube is one of the most attractive versions of the hypercube. It preserves the important features of the hypercube and reduces its diameter by a factor of two. This dissertation investigates relations and transformations between various interconnection networks and the twisted hypercube and explore its efficiency in parallel computation. The capability of the twisted hypercube to simulate complete binary trees, complete quad trees, and rings is demonstrated and compared with the hypercube. Finally, the fault-tolerance of the twisted hypercube is investigated. We present optimal algorithms to simulate rings in a faulty twisted hypercube environment and compare that with the hypercube

ONE BY ONE EMBEDDING THE CROSSED HYPERCUBE INTO PANCAKE GRAPH

Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from vertices of G to the vertices of H. The dilation of embedding is the maximum distance between f(u), f(v) taken over edges (u, v) of G. The Pancake graph is one as viable interconnection scheme for parallel computers, which has been examined by a number of researchers. The Pancake was proposed as alternatives to the hypercube for interconnecting processors in parallel computer. Some good attractive properties of this interconnection network include: vertex symmetry, small degree, a sub-logarithmic diameter, extendability, and high connectivity (robustness), easy routing and regularity of topology, fault tolerance, extensibility and embeddability of others topologies. In this paper, we give a construction of one by one embedding of dilation 5 of crossed hypercube into Pancake graph

The generalized 4-connectivity of burnt pancake graphs

The generalized $k$-connectivity of a graph $G$, denoted by $\kappa_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ and $|S|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. An $n$-dimensional burnt pancake graph $BP_n$ is a Cayley graph which posses many desirable properties. In this paper, we try to evaluate the reliability of $BP_n$ by investigating its generalized 4-connectivity. By introducing the notation of inclusive tree and by studying structural properties of $BP_n$, we show that $\kappa_4(BP_n)=n-1$ for $n\ge 2$, that is, for any four vertices in $BP_n$, there exist ($n-1$) internally edge disjoint trees connecting them in $BP_n$