A determinant formula for the Jones polynomial of pretzel knots

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above. Evaluations of these activity words are related to the chain complex for the Champanerkar-Kofman spanning tree model of reduced Khovanov homology.Comment: 19 pages, 12 figures, 2 table

Direct Freeform Fabrication of Spatially Heterogeneous Living Cell-Impregnated Implants

The objectives of this work are the development of the processes, materials, and tooling to directly “3-D print” living, pre-seeded, patient-specific implants of spatially heterogeneous compositions. The research presented herein attempts to overcome some of the challenges to scaffolding, such as the difficulty of producing spatially heterogeneous implants that require varied seeding densities and/or cell-type distributions. In the proposed approach, living implants are fabricated by the layer-wise deposition of pre-cell-seeded alginate hydrogel. Although alginate hydrogels have been previously used to mold living implants, the properties of the alginate formulations used for molding were not suitable for 3-D printing. In addition to changing the formulation to make the alginate hydrogels “printable,” we developed a robotic hydrogel deposition system and supporting CAD software to deposit the gel in arbitrary geometries. We demonstrated this technology’s capabilities by printing alginate gel implants of multiple materials with various spatial heterogeneities, including, implants with completely embedded material clusters. The process was determined to be both viable (94±5% n=15) and sterile (less than one bacterium per 0.9 µL after 8 days of incubation). Additionally, we demonstrated the printing of a meniscus cartilage-shaped gel generated directly from a CT Scan. The proposed approach may hold advantages over other tissue printing efforts [5,9]. This technology has the potential to overcome challenges to scaffolding and could enable the efficient fabrication of spatially heterogeneous, patient-specific, living implants.Mechanical Engineerin

Modeling the Internet's Large-Scale Topology

Network generators that capture the Internet's large-scale topology are crucial for the development of efficient routing protocols and modeling Internet traffic. Our ability to design realistic generators is limited by the incomplete understanding of the fundamental driving forces that affect the Internet's evolution. By combining the most extensive data on the time evolution, topology and physical layout of the Internet, we identify the universal mechanisms that shape the Internet's router and autonomous system level topology. We find that the physical layout of nodes form a fractal set, determined by population density patterns around the globe. The placement of links is driven by competition between preferential attachment and linear distance dependence, a marked departure from the currently employed exponential laws. The universal parameters that we extract significantly restrict the class of potentially correct Internet models, and indicate that the networks created by all available topology generators are significantly different from the Internet

Adiabatic Elimination in a Lambda System

This paper deals with different ways to extract the effective two-dimensional lower level dynamics of a lambda system excited by off-resonant laser beams. We present a commonly used procedure for elimination of the upper level, and we show that it may lead to ambiguous results. To overcome this problem and better understand the applicability conditions of this scheme, we review two rigorous methods which allow us both to derive an unambiguous effective two-level Hamiltonian of the system and to quantify the accuracy of the approximation achieved: the first one relies on the exact solution of the Schrodinger equation, while the second one resorts to the Green's function formalism and the Feshbach projection operator technique.Comment: 14 pages, 3 figure

Charge Transfer in Partition Theory

The recently proposed Partition Theory (PT) [J.Phys.Chem.A 111, 2229 (2007)] is illustrated on a simple one-dimensional model of a heteronuclear diatomic molecule. It is shown that a sharp definition for the charge of molecular fragments emerges from PT, and that the ensuing population analysis can be used to study how charge redistributes during dissociation and the implications of that redistribution for the dipole moment. Interpreting small differences between the isolated parts' ionization potentials as due to environmental inhomogeneities, we gain insight into how electron localization takes place in H2+ as the molecule dissociates. Furthermore, by studying the preservation of the shapes of the parts as different parameters of the model are varied, we address the issue of transferability of the parts. We find good transferability within the chemically meaningful parameter regime, raising hopes that PT will prove useful in chemical applications.Comment: 12 pages, 16 figure