Incompressible Surfaces in Almost Alternating Link Exteriors

In this paper, we discuss mainly the properties of incompressible pairwise incomprcssiblc surfaccs in ahnost altcrnating link complcmcnts. Lct L bc a almost link and lct F be an incompressible pairwise incompressible surface in S^3 - L. First, we give the properties that the surface F intersects with 2-spheres in S^3- L. The intersection consisting of a collection of circles and saddle-shaped discs is called a topological graph. One can compute the Euler Characteristic number of the surface by calculating the characteristic number of the graph. Next, we prove that if the graph is special simple, then the genus of the surface is zero.

Alternating link and its generalization

The alternating links give a classical class of links.They play an important role in Knot Theory.Ozsvath and Szab6 introduced a quasi-alternating link which is a generalization of an alternating link.In this paper we review some results of alternating links and quasi-alternating links on the Jones polynomial and the Khovanov homology.Moreover,we introduce a long pass link.Several problems worthy of further study are provided.

INCOMPRESSIBLE PAIRWISE INCOMPRESSIBLE SURFACES IN LINK COMPLEMENTS

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S3 - L. We discuss the properties that the surface F intersects with 2-spheres in S3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves （R-move and S2-move）, and define the characteristic number of the topological graph for F∩S2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating （or almost alternating） link complements. We prove that the genus of the surface equals zero if the component number of F∩S2＋（or F∩S2-） is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #（F） ≤8.

Central Circuit Coverings of Octahedrites and Medial Polyhedra

An octahedrite is a 4-valent polyhedron with only 3-faces and 4-faces. We study edge-partitions of some octahedrites, medial and related polyhedra into central circuits.

The Number of Circles of a Maximum State of a Plane Graph with Applications

Motivated by the connection with the genus of the corresponding link and its application on DNA polyhedral links,in this paper,we introduce a parameter s_(max)(G),which is the maximum number of circles of states of the link diagram D(G)corresponding to a plane(positive)graph G.We show that s_(max)(G)does not depend on the embedding of G and if G is a 4-edge-connected plane graph then s_(max)(G)is equal to the number of faces of G,which cover the results of S.Y.Liu and H.P.Zhang as special cases.

Modeling and Analysis of PV Configurations to Extract Maximum Power Under Partial Shading Conditions

Photovoltaics(PV)are widely used as renewable energy sources for standalone and grid connected PV systems.But these PV systems face major reduction in output power and efficiency due to Partial Shading Conditions(PSCs).This research paper focuses on the different choices of optimum PV Configuration under a given shading pattern to extract maximum power by mitigating mismatching loss.Various PV configurations,such as Series(S),Series Parallel(SP),Total Cross Tied(TCT),Bridge Linked(BL),Honey Comb(HC)and Alternate Total Cross Tied–Bridge Linked(A-TCT-BL)are modeled and analyzed under PSCs.Nine shading patterns,such as center,diagonal,corner,L-shaped,short and narrow,short and wide,long and narrow,long and wide and random,are considered to study the behavior of a 6×6 array form of a PV Configuration.Their performances are compared based on open circuit voltage,short circuit current,global maximum power point(GMPP),maximum voltage,maximum current,shading loss,fill factor,mismatching loss and efficiency.A novel Hybrid Configuration called A-TCT-BL PV Configuration is proposed to generate maximum power under PSCs and to minimize the number of cross ties and wiring complexities.This Configuration is an integration of TCT and BL PV Configuration and the simulation results prove the capability of this proposed PV Configuration to generate maximum power,fill factor,efficiency and minimum mismatching loss compared to S,SP,BL and HC PV Configurations under a majority of the PSCs investigated.A Canadian Solar CS5P-200M PV module is considered for simulation and is simulated using Matlab/Simulink software.