Topological Genus Research Articles

Automorphism groups of bordered non-orientable Klein surfaces of topological genus 4

Automorphism groups of bordered non-orientable Klein surfaces of topological genus 4

The full group of automorphisms of non-orientable unbordered Klein surfaces of topological genus 8

It is very difficult to determine the full automorphism group of compact Riemann and Klein surfaces. The automorphism groups of hyperelliptic surfaces, and those of (compact, non-orientable, unbordered) Klein surfaces whose genus is less or equal to 7 are known, but a lot of hard work is left to do. To amplify the knowledge in this field, in this paper the full automorphism groups of the surfaces of genus 8 are calculated.

Scaling between elasticity and topological genus for random network nanomaterials

We explore the hypothesis that the variation of the effective, macroscopic Young’s modulus, Eeff, of a random network material with its scaled topological genus, g, and with the solid fraction, φ, can be decomposed into the product of g- and φ-dependent functions. Based on findings for nanoporous gold, supplemented by the Gibson–Ashby scaling law for Eeff, we argue that both functions are quadratic in bending-dominated structures. We present finite-element-modeling results for Eeff of coarsened microstructures, in which g and φ are decoupled. These results support the quadratic forms.

Groups of automorphisms of bordered non-orientable Klein surfaces of topological genus 3

In this paper we study the automorphism groups G of compact non-orientable Klein surfaces of topological genus 3 with k>0 boundary components. We obtain that the group G is one of the following: C2, C3, C4, C6, C2×C2, D3, D4 and D6. Besides, for each k>0 there exists such a Klein surface admitting each of these groups as an automorphism group, excepting that C6 and D6 do not act on surfaces with k≡1mod3.Moreover, if G is an automorphism group of a bordered non-orientable Klein surface of topological genus 3, with k boundary components, then it is the full automorphism group of some such surface excepting C3 for k=1 and 2, C4 for k=1, and C6 for k=2 and 5.

Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layout

G-splines are a generalization of B-splines that deals with extraordinary points by imposing G1 constraints across their spoke edges, thus obtaining a continuous tangent plane throughout the surface. Using the isoparametric concept and the Bubnov–Galerkin method to solve partial differential equations with G-splines results in discretizations with global C1 continuity in physical space. Extraordinary points (EPs) are required to represent manifold surfaces with arbitrary topological genus. In this work, we allow both interior and boundary EPs and there are no limitations regarding how close EPs can be from each other. Reaching this level of flexibility is necessary so that splines with EPs can become mainstream in the design-through-analysis cycle of the complex thin-walled structures that appear in engineering applications. To the authors’ knowledge, the two EP constructions based on imposing G1 constraints proposed in this work are the first two EP constructions used in isogeometric analysis (IGA) that combine the following distinctive characteristics: (1) Only vertex-based control points are used and they behave as geometric shape handles, (2) any control point of the control net can potentially be an EP, (3) global C1 continuity in physical space is obtained without introducing singularities, (4) faces around EPs are not split into multiple elements, i.e., Bézier meshes with uniform element size are obtained, and (5) good surface quality is attained. The studies of convergence and surface quality performed in this paper suggest that G-splines are more suitable for IGA than EP constructions based on the D-patch framework. Finally, we have represented the stiffener, the inner part, and the outer part of a B-pillar with G-spline surfaces and solved eigenvalue problems using both Kirchhoff–Love and Reissner–Mindlin shell theories. The results are compared with bilinear quadrilateral meshes and excellent agreement is found between G-splines and conventional finite elements. In summary, G-splines are a viable alternative to design and analyze thin-walled structures using the same geometric representation so as to streamline the design-through-analysis cycle.

The topological slice genus of satellite knots

This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $\mathbb{Z}$-slice genus. As an application, we show that the $(n,1)$-cable of any 3-genus 1 knot (e.g. the figure 8 or trefoil knot) has topological slice genus at most 1. Further, we show that the lower bounds on the slice genus coming from the Tristram-Levine and Casson-Gordon signatures cannot be used to disprove the conjecture. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern $P$. This stands in stark contrast with the smooth category, where for example there are many genus 1 knots whose $(n,1)$-cables have arbitrarily large smooth 4-genera.

P-Groups of automorphisms of compact non-orientable Riemann surfaces

We study p-groups of automorphisms of compact non-orientable Riemann surfaces of topological genus gge 3. We obtain upper bounds of the order of such groups in terms of p, g and the minimal number of generators of the group. We also determine those values of g for which these bounds are sharp. Furthermore, the same kind of results are obtained when the p-group acts as the full automorphism group of the surface.

4-manifolds and topological modular forms

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1, 0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0, 1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on ’t Hooft anomalies of 6d (1, 0) theories and a better understanding of the relation between 2d (0, 1) theories and TMF spectra.

Regarding the Euler–Plateau problem with elastic modulus

We study equilibrium configurations for the Euler–Plateau energy with elastic modulus, which couples an energy functional of Euler–Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is shown that the potential minimizers of this energy are highly dependent on the choice of physical rigidity parameters, and that the area of critical surfaces can be computed entirely from their boundary data. When the elastic modulus does not vanish, it is shown that axially symmetric critical immersions and critical immersions of disk type are necessarily planar domains bounded by area-constrained elasticae. The cases of topological genus zero with multiple boundary components and unrestricted genus with control on the geodesic torsion are also discussed, and sufficient conditions are given which establish the same conclusion in these cases.

A lower bound for the double slice genus

In this paper, we develop a lower bound for the double slice genus of a knot in the 3-sphere using Casson-Gordon invariants. As an application, we show that the difference between the slice genus and the double slice genus can be arbitrarily large. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound. In particular, through the study of superslice genus we show that the degree of the Alexander polynomial is an upper bound for the topological double slice genus of a ribbon knot.

Determining Genus From Sandpile Torsor Algorithms

We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.

String Perturbation Theory on the Schwinger-Keldysh Time Contour.

We perform the large-N expansion in the Schwinger-Keldysh formulation of nonequilibrium quantum systems with matrix degrees of freedom and study universal features of the anticipated dual string theory. We find a rich refinement of the topological genus expansion: In the original formulation, the future time instant where the forward and backward branches of the Schwinger-Keldysh time contour meet is associated with its own world sheet genus expansion. After the Keldysh rotation, the world sheets decompose into a classical and quantum part.

Turning a Superhydrophilic Surface Weakly Hydrophilic: Topological Wetting States.

For water droplets placed on a rough or structured surface, two distinct wetting states commonly observed are either the Wenzel state (droplets wet the surface without showing air pockets beneath the droplets) or the Cassie state (droplets reside on top of the structure with air pockets trapped beneath the droplets). Herein, we show molecular dynamics (MD) simulation evidence of a previously unreported wetting behavior, i.e., the rise of multiple Wenzel states on the structured surfaces whose flat-surface counterparts are superhydrophilic (i.e., complete wetting surfaces with the hallmark of zero contact angle for water droplets). Specifically, our MD simulations show that on the structured surfaces with topology of closed-loop nanowalls/nanochannels, the water droplet can exhibit multiple Wenzel wetting states with the apparent contact angles >0°. We name these distinct multiple Wenzel states as "topological wetting states" because their existence can be attributed to the topology of the closed-loop nanowalls/nanochannels. Regardless of the shape of the closed loops, such topological wetting states can always arise due to the topological invariant (i.e., all closed loops entail the same topological genus value). This unusual wetting behavior is contrary to the conventional view (and to the prediction of the Wenzel equation), namely, a rough hydrophilic surface should have stronger hydrophilicity than its flat-surface counterpart.

The topological symmetric orbifold

We analyze topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the cohomology of the Hilbert scheme on the boundary.

Textile-Based Capacitive Sensor for Physical Rehabilitation via Surface Topological Modification.

Wearable sensor technologies, especially continuous monitoring of various human health conditions, are attracting increased attention. However, current rigid sensors present obvious drawbacks, like lower durability and poor comfort. Here, a strategy is proposed to efficiently yield wearable sensors using cotton fabric as an essential component, and conductive materials conformally coat onto the cotton fibers, leading to a highly electrically conductive interconnecting network. To improve the conductivity and durability of conductive coatings, a topographical modification approach is developed with genus-3 and genus-5 structures, and topological genus structures enable cage metallic seeds on the surface of substrates. A textile-based capacitive sensor with flexible, comfortable, and durable properties has been demonstrated. High sensitivity and convenience of signal collection have been achieved by the excellent electrical conductivity of this sensor. Based on results of deep investigation on capacitance, effects of distance and angles between two conductive fabrics contribute to the capacitive sensitivity. In addition, the textile-based capacitive sensor has successfully been used for real-time monitoring human breathing, speaking, blinking, and joint motions during physical rehabilitation exercises.

Signature and concordance of virtual knots

We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and $\omega$-signatures are shown to give bounds on the topological slice genus of almost classical knots, and they are applied to address a recent question of Dye, Kaestner, and Kauffman on the virtual slice genus of classical knots. A conjecture on the topological slice genus is formulated and confirmed for all classical knots with up to 11 crossings and for 2150 out of 2175 of the 12 crossing knots. The Seifert pairing is used to define directed Alexander polynomials, which we show satisfy a Fox-Milnor criterion when the almost classical knot is slice. We introduce virtual disk-band surfaces and use them to establish realization theorems for Seifert matrices of almost classical knots. As a consequence, we deduce that any integral polynomial $\Delta(t)$ satisfying $\Delta(1)=1$ occurs as the Alexander polynomial of an almost classical knot. In the last section, we use parity projection and Turaev's coverings of knots to extend the Tristram-Levine signatures to all virtual knots. A key step is a theorem saying that parity projection preserves concordance of virtual knots. This theorem implies that the signatures, $\omega$-signatures, and Fox-Milnor criterion can be lifted to give slice obstructions for all virtual knots. There are 76 almost classical knots with up to six crossings, and we use our invariants to determine the slice status for all of them and the slice genus for all but four. Table 2 at the end summarizes our findings.

Seamless integration of design and Kirchhoff–Love shell analysis using analysis-suitable unstructured T-splines

Analysis-suitable T-splines (ASTS) including both extraordinary points and T-junctions are used to solve Kirchhoff–Love shell problems. Extraordinary points are required to represent surfaces with arbitrary topological genus. T-junctions enable local refinement of regions where increased resolution is needed. The benefits of using ASTS to define shell geometries are at least two-fold: (1) The manual and time-consuming task of building a new mesh from scratch using the CAD geometry as an input is avoided and (2) C1 or higher inter-element continuity enables the discretization of shell formulations in primal form defined by fourth-order partial differential equations. A complete and state-of-the-art description of the development of ASTS, including extraordinary points and T-junctions, is presented. In particular, we improve the construction of C1-continuous non-negative spline basis functions near extraordinary points to obtain optimal convergence rates with respect to the square root of the number of degrees of freedom when solving linear elliptic problems. The applicability of the proposed technology to shell analysis is exemplified by performing geometrically nonlinear Kirchhoff–Love shell simulations of a pinched hemisphere, an oil sump of a car, a pipe junction, and a B-pillar of a car with 15 holes. Building ASTS for these examples involves using T-junctions and extraordinary points with valences 3, 5, and 6, which often suffice for the design of free-form surfaces. Our analysis results are compared with data from the literature using either a seven-parameter shell formulation or Kirchhoff–Love shells. We have also imported both finite element meshes and ASTS meshes into the commercial software LS-DYNA, used Reissner–Mindlin shells, and compared the result with our Kirchhoff–Love shell results. Excellent agreement is found in all cases. The complexity of the shell geometries considered in this paper shows that ASTS are applicable to real-world industrial problems.

Topology evolution during coarsening of nanoscale metal network structures

Many experiments exploit curvature-driven, surface-diffusion-mediated coarsening for tuning the characteristic structure size of metal network structures made by dealloying, such as nanoporous gold. Here we study this process by kinetic Monte Carlo simulation. The initial microstructures are leveled Gaussian random fields, approximating spinodally decomposed mixtures, of different solid fraction $\ensuremath{\varphi}$. Earlier work establishes these structures as valid representations of the nanoporous gold microstructure. We find that the coarsening law for the characteristic spacing between the ligaments of the network is universal, whereas the time evolution of the characteristic ligament diameter is not. The expected time exponent 1/4 is confirmed by our simulation. Contrary to what may be expected based on continuum models, the degree of surface faceting or roughness has no apparent effect on the coarsening kinetics. In the time interval of our study, the network connectivity---as measured by a scaled density of topological genus---remains sensibly invariant for networks with $\ensuremath{\varphi}\ensuremath{\ge}0.3$, consistent with previous reports of a self-similar evolution of the microstructure during coarsening. Yet, networks with lesser $\ensuremath{\varphi}$ lose their connectivity on coarsening and can even undergo a percolation-to-cluster transition. This process is slow for $\ensuremath{\varphi}$ only little below 0.3 and it accelerates in networks with lesser $\ensuremath{\varphi}$. The dependency of the connectivity evolution on $\ensuremath{\varphi}$ may explain controversial findings on the microstructure evolution of nanoporous gold in experimental studies.

The block spectrum of RNA pseudoknot structures.

In this paper we analyze the length-spectrum of blocks in [Formula: see text]-structures. [Formula: see text]-structures are a class of RNA pseudoknot structures that play a key role in the context of polynomial time RNA folding. A [Formula: see text]-structure is constructed by nesting and concatenating specific building components having topological genus at most [Formula: see text]. A block is a substructure enclosed by crossing maximal arcs with respect to the partial order induced by nesting. We show that, in uniformly generated [Formula: see text]-structures, there is a significant gap in this length-spectrum, i.e., there asymptotically almost surely exists a unique longest block of length at least [Formula: see text] and that with high probability any other block has finite length. For fixed [Formula: see text], we prove that the length of the complement of the longest block converges to a discrete limit law, and that the distribution of short blocks of given length tends to a negative binomial distribution in the limit of long sequences. We refine this analysis to the length spectrum of blocks of specific pseudoknot types, such as H-type and kissing hairpins. Our results generalize the rainbow spectrum on secondary structures by the first and third authors and are being put into context with the structural prediction of long non-coding RNAs.

On classical upper bounds for slice genera

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.