Checkerboard Surface Research Articles

Right-angled links in thickened surfaces

Traditionally, alternating links are studied with alternating diagrams on S2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^2$$\\end{document} in S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^3$$\\end{document}. In this paper, we consider links which are alternating on higher genus surfaces F in thickened surface F×I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F \ imes I$$\\end{document}. We define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property is equivalent to the link having two totally geodesic checkerboard surfaces, equivalent to each checkerboard surface consisting of one type of polygon, and equivalent to a set of restrictions on the link’s alternating projection diagram. We then use these diagram restrictions to classify RGCR links according to the polygons in their checkerboard surfaces, provide a bound on the number of RGCR links for a given surface of genus g, and find a right-angled knot in a thickened surface. Along the way, we answer a question posed by Champanerkar, Kofman, and Purcell about links with alternating projections on the torus.

Simplistic metasurface design approach for incident angle and polarization insensitive rcs reduction

This paper proposes the design of a metasurface for polarization and incident angle-insensitive RCS reduction applications. An ellipse-shaped unit cell is utilized as a polarization converter, which is then arranged to form a checkerboard surface. While a single layer checkerboard structure gives a wideband RCS reduction, a double layer structure yields polarization and incident angle independent operation. The two layers have unit cells rotated 450 to each other. Experimental results demonstrate an RCS reduction bandwidth of around 90%. Further the RCS reduction remains stable with polarization and incident angle variation.

Enhanced liquid replenishment for pool boiling heat transfer and its CHF mechanism on patterned freeze-casted surfaces

In the current investigation, the pool boiling heat transfer process of patterned porous surfaces, as well as a smooth surface, in ethanol was examined from an academic perspective. The freeze casting (also known as ice-templating) processes was used to fabricate the structured surfaces. It was discovered that freeze-casted coatings significantly improved the performance of boiling heat transfer in comparison to the smooth surface (SMS). The most remarkable increase in Critical Heat Flux (CHF) was achieved by checkerboard porous surface(CHECKERBOARD), with an improvement of 81.6% relative to that of SMS. The maximum Heat Transfer Coefficient (HTC) enhancement was obtained by full porous surface (FULLY-COVERED), with an HTC value 215.8% higher than that of the SMS. For the semicircular porous surface (SEMICIRCULAR), striped porous surface (STRIPED), annular porous surface (ANNULAR) and checkerboard porous surface(CHECKERBOARD) with 50% coating coverage, CHECKERBOARD demonstrated the best heat transfer performance, with CHF and HTC increased by 32.3% and 48.9%, respectively, compared to the SEMICIRCULAR surface. The mechanism of liquid supply at the CHF of the patterned porous surfaces was investigated through phenomenological observations of the boiling phenomena. A modified model, accounting for the coalesced bubble departure frequency and capillary wicking effects, was proposed for CHF prediction. It is noteworthy that the fluid replenishment considers both vertical and lateral replenishment of porous coatings. The CHF data from this study, along with existing literature, were utilized to validate the model, and the predicted results exhibited good agreement with the experimental data, with errors within ±8%.

A Low-Profile Multifunctional Metasurface Reflector for Multiband Polarization Transformation

This brief proposes a low-profile multifunctional metasurface reflector for simultaneous linear to linear (LP-to-LP) and linear to circular (LP-to-CP) polarization conversion over multiple bands. The metasurface unit cell comprises a split ring resonator-loaded cross-diagonal pattern printed at the top of a dielectric substrate and a thin air space that separates the metallic bottom plate. The proposed reflector exhibits tri-band LP-to-LP conversion at 5.47-5.68, 7.86-8.84, 14.68-16.83 GHz, and quad-band LP-to-CP conversion at 5.30-5.41, 5.77-7.58, 9.27-13.91, 17.53-19.59 GHz frequency range. The response of the surface is analyzed through an equivalent circuit model (ECM), which gives physical insight into the metasurface. Moreover, the metasurface is explored through the characteristics mode (CM), where significant modes are determined based on the modal behaviors. Thereafter, modal behaviors are further investigated for simultaneous linear and circular polarization. Furthermore, the proposed structure and its mirror symmetric configuration are employed to develop the checkerboard surface. Such surface essentially realizes the phase gradient mechanism, which results in radar cross-section (RCS) reduction in the frequency range of 5.85-5.95, 9.2-10.3, and 16.6-18.5 GHz. The measured results are in good agreement with the simulated response.

Simultaneous Beam Forming and Focusing Using a Checkerboard Anisotropic Surface

A novel design method of simultaneous beam forming and focusing using a checkerboard anisotropic surface is proposed and verified in this paper. The proposed multibeam control regardless of far and near regions can easily be achieved through a rearrangement of the checkerboard structure. The unit cell of the utilized anisotropic surface consists of two identical metallic structures divided by a dielectric material. When the EM wave with a circular polarization (CP) is incident on the unit cell, the maximum transmission phase variation of the unit cell is 360 degrees by half rotation of the unit cell. A microstrip patch antenna with trimmed corners is used to launch the CP wave and the distance between the microstrip patch antenna and anisotropic surface is about 2 wavelengths considering the optimized spillover and taper efficiencies. After designing each anisotropic surface for beam forming and focusing, the unit cells of the surface are rearranged in the form of a checkerboard. The feasibility of the proposed method is confirmed by full-wave simulation and measurement for anisotropic surface with a beam forming angle of 30 degrees and beam focusing point 60 mm away from center at 5.8 GHz. The forming angle and focal length are simulated and measured to be 28 degrees and about 65 mm, respectively.

The Jones polynomial from a Goeritz matrix

AbstractWe give an explicit algorithm for calculating the Kauffman bracket of a link diagram from a Goeritz matrix for that link. Further, we show how the Jones polynomial can be recovered from a Goeritz matrix when the corresponding checkerboard surface is orientable, or when more information is known about its Gordon–Litherland form. In the process we develop a theory of Goeritz matrices for cographic matroids, which extends the bracket polynomial to any symmetric integer matrix. We place this work in the context of links in thickened surfaces.

Shape-Controlled Pathways in the Hydrogen Production from Ethanol Steam Reforming over Ceria Nanoparticles.

The ethanol surface reaction over CeO2 nanooctahedra (NO) and nanocubes (NC), which mainly expose (111) and (100) surfaces, respectively, was studied by means of infrared spectroscopy (TPSR-IR), mass spectrometry (TPSR-MS), and density functional theory (DFT) calculations. TPSR-MS results show that the production of H2 is 2.4 times higher on CeO2-NC than on CeO2-NO, which is rationalized starting from the different types of adsorbed ethoxy species controlled by the shape of the ceria particles. Over the CeO2(111) surface, monodentate type I and II ethoxy species with the alkyl chain perpendicular or parallel to the surface, respectively, were identified. Meanwhile, on the CeO2(100) surface, bidentate and monodentate type III ethoxy species on the checkerboard O-terminated surface and on a pyramid of the reconstructed (100) surface, respectively, are found. The more labile surface ethoxy species on each ceria nanoshape, which are the monodentate type I or III ethoxy on CeO2-NO and CeO2-NC, respectively, react on the surface to give acetate species that decompose to CO2 and CH4, while H2 is formed via the recombination of hydroxyl species. In addition, the more stable monodentate type II and bidentate ethoxy species on CeO2-NO and CeO2-NC, respectively, give an ethylenedioxy intermediate, the binding of which is facet-dependent. On the (111) facet, the less strongly bound ethylenedioxy desorbs as ethylene, whereas on the (100) facet, the more strongly bound intermediate also produces CO2 and H2 via formate species. Thus, on the (100) facet, an additional pathway toward H2 formation is found. ESR activity measurements show an enhanced H2 production on the nanocubes.

Ripple Patterns Spontaneously Emerge through Sequential Wrinkling Interference in Polymer Bilayers.

We report the formation of "ripple" patterns by the sequential superposition of nonorthogonal surface waves excited by the spontaneous buckling of polymeric bilayers. Albeit of a different nature and micron scale compared to the familiar sedimentary ripples caused by gentle wave oscillations, we find commonalities in their topography, defects, and bifurcations. The patterns are rationalized in terms of a defect density that depends on the relative angle between generations, and a constant in-plane bending angle that depends on skin thickness. A minimal wave summation model enables the design of ripple and checkerboard surfaces by tuning material properties and fabrication process.

Alternating links with totally geodesic checkerboard surfaces

We prove that alternating links with two totally geodesic checkerboard surfaces are three links with projection the 1-skeleton of the octahedron, the cuboctahedron and the icosidodecahedron. We also characterize these links as right-angled completely realisable links and show that all hyperbolic weaving knots with two exceptions have both checkerboard surfaces not totally geodesic.

A switchable polarisation conversion/in‐phase reflection metasurface and its application

A switchable polarisation conversion/in-phase reflection metasurface (PCIPR-MS) based on a Pin diode is proposed in this study. By changing the working state of the Pin diode, the PCIPR-MS can be used as a polarisation conversion metasurface (PCM) and an artificial magnetic conductor (AMC), respectively. Furthermore, the designed PCIPR-MS can be arranged into a checkerboard surface to integrate with the antenna, which means that the PCIPR-MS can be flexibly switched to control the radiation and scattering characteristics of the antenna in a time-sharing manner. The simulation results show that the 3-dB axial ratio (AR) bandwidth is 4.31 to 6.14 GHz (35%) with the application of the AMC as the metal ground plane of the antenna in the radiation state, and the right-hand circularly polarised (RHCP) gain is enhanced by more than 1.5 dB from 4.65 to 5.80 GHz in a low-profile compared with the reference antennas. By using the PCM, phase cancelation is achieved in the scattering state of the antenna to realise the low-RCS characteristics.

State graphs and fibered state surfaces

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graphs, including those associated with 2-bridge links. This characterizes fibering for many families of state surfaces.

Scalable Manufacturing of Bending-Induced Surface Wrinkles.

The wide range of textures that can be generated via wrinkling can imbue surfaces with functionalities useful for a variety of applications including tunable optics, stretchable electronics, and coatings with controlled wettability and adhesion. Conventional methods of wrinkle fabrication rely on batch processes in piece-by-piece fashion, not amenable for scale-up to enable commercialization of surface wrinkle-related technologies. In this work, a scalable manufacturing method for surface wrinkles is demonstrated on a cylindrical support using bending-induced strains. A bending strain is introduced to a thin layer of ultraviolet-curable poly(dimethylsiloxane) (UV-PDMS) coated on top of a soft PDMS substrate by wrapping the bilayer around a cylindrical roller. After curing the UV-PDMS and subsequently releasing the bending strain, one-dimensional or checkerboard surface wrinkles are produced. Based on experimental and computational analyses, we show that these patterns form as a result of the interplay between swelling and bending strains. The feasibility of continuous manufacturing of surface wrinkles is demonstrated by using a two-roller roll-to-roll prototype, which paves the way for scalable roll-to-roll processing. To demonstrate the utility of these textures, we show that surface wrinkles produced in this manner enhance the light harvesting and thus efficiency of a solar cell at oblique angles of illumination due to their strong light scattering properties.

Geometry of alternating links on surfaces

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude exceptional Dehn fillings.

Design of Checkerboard AMC Structure for Wideband RCS Reduction

In this paper, we propose a method for designing checkerboard surfaces using strategically placed artificial magnetic conductors (AMCs) for wideband radar cross section (RCS) reduction. The theoretical analysis shows that the reflection phases of the two AMCs should be linear. The cancelation conditions of checkerboard structure are analytically derived using an equivalent transmission line model, and the ideal impedance conditions of the two AMCs are obtained, which serve as guidance to select patterns of two AMCs unit. A planar checkerboard surface with two kinds of AMC elements is designed for wideband RCS reduction. A 10 dB RCS reduction is observed for approximately 91.5% of the relative bandwidth (3.77–10.14 GHz) under normal incidence. A prototype of the proposed checkerboard surface is fabricated, and the measurements results show good coincidence with the synthetic studies. The bistatic RCS of the checkerboard surface is considered for both transverse electric and transverse magnetic polarizations under oblique incidence.

Geometry of biperiodic alternating links

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links.

An Optimized Checkerboard Structure for Cross-Section Reduction: Producing a Coating Surface for Bistatic Radar Using the Equivalent Electric Circuit Model

In this article, checkerboard metallic surfaces are analytically studied for nonabsorptive radar cross-section (RCS) reduction. The patch periodic elements are modeled with the equivalent electric circuit. The circuit model is exploited to achieve an improved reflection phase from the unit cell. By employing the phase difference from the edges of two frequencyselective surfaces (FSSs), a destructive interference takes place. In the wavescattering scenario, a fitness function is derived for RCS reduction (RCSR). The optimum reflection phase for an incident wave frequency interval is obtained with the help of circuit model analysis. A genetic algorithm (GA) is used for reflection phase optimization. An optimum checkerboard is designed based on the analytical and optimization methods. The checkerboard is constructed with two different patch widths as the optimum board to realize a phase difference condition in two dimensions. The designed board is numerically studied for both polarizations, bistatic RCS, and an incident angle span. After evaluating the numerical solution, the designed checkerboard surface is fabricated. Experimental observation validates the analytical results. The studied board uses a wellknown periodic surface and has a concrete analytical basis, optimum design procedure, bistatic RCSR, and far-field investigation.

An efficient reflective polarization rotation meta surface for broadband RCS reduction

A compact and broadband reflective polarization rotation meta surface (PRMS) with high polarization conversion ratio (PCR) and radar cross section (RCS) reduction is proposed. A linearly polarized incident electromagnetic (EM) wave is reflected with 90° polarization rotation using the proposed PRMS. The reflective PRMS unit cell consists of complementary stacked quasi fractal patch along with logarithmic stepped patch. This reflective PRMS is printed on stacked FR4 substrates covered by FR4 superstrate and is backed by a ground plane. The patches are connected to the metallic ground using non symmetric metallic vias and the overall dimension of the unit cell in terms of its minimum operating wavelength is only 0.088λ0 × 0.088λ0 × 0.082λ0. A polarization bandwidth of 97.69% with a maximum PCR of 99% is achieved. Further, the proposed reflective PRMS is arranged periodically in a checkerboard surface to obtain a broadband RCS reduction of 10 dB in the required band. A prototype of the checkerboard surface is fabricated and tested for the validation of simulation results. A good pact is obtained between the simulation and experimental results.

Reducing reflection of bandpass frequency selective surface using checkerboard surface

A hybrid structure with the properties of low-frequency transmission and reflection reduction at high frequencies is designed and demonstrated experimentally. The bandpass frequency selective surface (FSS) working at low frequencies is firstly proposed. Then, the checkerboard surface that can scatter the reflected electromagnetic (EM) wave is also introduced on the top layer. The two layers are separated by a foam spacer to gain the hybrid structure. Due to the metal-like reflection behavior of the FSS at high frequencies, integrating with the checkerboard surface results in low reflection in a broad frequency band, while the efficient transmission at low frequencies can be achieved all the same. Simulated results show that the proposed hybrid structure can achieve a reflection reduction of more than 10 dB in the frequency range of 8.4–18 GHz, as well as the transmission coefficient greater than −1 dB from 3.7 to 4.3 GHz. Finally, the experimental measurement has also been performed to provide a good demonstration. The proposed results can be seen as a good candidate in the applications of stealth techniques, EM compatibility and so on.

Cylindrically Curved Checkerboard Surfaces for Radar Cross-Section Reduction

Checkerboard surfaces, for radar cross-section (RCS) reduction, utilizing artificial magnetic conductor structures on flexible cylindrically curved ground planes are introduced. The RCSs of cylindrical checkerboard surfaces are examined for two different radii of curvature. Wideband curved checkerboard surfaces are evaluated under normal incidence for HH and VV polarizations. Simulated bistatic RCS patterns of the cylindrical checkerboard surfaces are presented, discussed, and justified, and the backscattering is compared with measurements. A very good agreement is observed.

High‐gain low‐RCS slot antenna array based on checkerboard surface

In-band and out-of-band radar cross-section reduction (RCSR) as well as gain enhancement for a slot antenna array is presented. The checkerboard surface (CS), combining an electromagnetic band gap structure and a metallic grounded slab, is adopted to realise a wideband RCSR. The analysis indicates a 10 dB RCSR of the CS compared with the perfect electric conductor surface can be achieved over 60% frequency bandwidth. Then, the designed CS is placed on the top surface of the 2 × 2 slot antenna array, and the measured results reveal that the proposed slot array possesses in-band and out-of-band RCSR from 5 to >12 GHz for both co- and cross-polarisations. Apart from the wideband RCSR, the gain enhancement is about 2.1 dB compared with the slot antenna array without the CS.