topological characterization
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Author(s):  
Philip Smith ◽  
Eleni Panagiotou

Abstract Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in 3-space. In this manuscript, we examine the topological complexity of polygonal chains in 3-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in 3-space and by uniform random walks in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For uniform random walks in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as $O(n^2)$ with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in 3-space increases as $O(n)$. These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement.


2021 ◽  
Vol 31 (2) ◽  
pp. 145-161
Author(s):  
Shibsankar Das ◽  
◽  
Shikha Rai ◽  

A topological index is a numerical quantity that defines a chemical descriptor to report several physical, biological and chemical properties of a chemical structure. In recent literature, various degree-based topological indices of a molecular structure are easily calculated by deriving a M-polynomial of that structure. In this paper, we first determine the expression of a M-polynomial of the triangular Hex-derived network of type three of dimension n and then obtain the corresponding degree-based topological indices from the closed form of M-polynomial. In addition, we use Maple software to represent the M-polynomial and the concerned degree-based topological indices pictorially for different dimensions.


Author(s):  
Hans U. Boden ◽  
Homayun Karimi

We use an extension of Gordon–Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If $\Sigma$ is a closed oriented surface and $F$ is a compact unoriented surface in $\Sigma \times I$ , then the Gordon–Litherland pairing defines a symmetric bilinear pairing on the first homology of $F$ . A compact surface in $\Sigma \times I$ is called definite if its Gordon–Litherland pairing is a definite form. We prove that a link $L$ in a thickened surface is non-split, alternating, and of minimal genus if and only if it bounds two definite surfaces of opposite sign.


2021 ◽  
Author(s):  
Ching Hua Lee ◽  
Ruizhe Shen

Abstract Strong, non-perturbative interactions often lead to new exciting physics, as epitomized by emergent anyons from the Fractional Quantum hall effect. Within the actively investigated domain of non-Hermitian physics, we discover a new family of states known as non-Hermitian skin clusters. Taking distinct forms as Vertex, Topological, Interface, Extended and Localized skin clusters, they generically originate from asymmetric correlated hoppings on a lattice, in the strongly interacting limit with quenched single-body energetics. Distinct from non-Hermitian skin modes which accumulate at boundaries, our skin clusters are predominantly translation invariant particle clusters. As purely interacting phenomena, they fall outside the purview of generalized Brillouin zone analysis, although our effective lattice formulation provides alternative analytic and topological characterization. Non-Hermitian skin clusters fundamentally originate from the fragmentation structure of the Hilbert space, and may thus be of significant interest in modern many-body contexts like the ETH and quantum scars.


Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 1991
Author(s):  
Muhammad Asif ◽  
Bartłomiej Kizielewicz ◽  
Atiq ur Rehman ◽  
Muhammad Hussain ◽  
Wojciech Sałabun

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θϕ formed by ϕ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θϕΩ. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of θϕΩ to enable its emerging use.


2021 ◽  
pp. 1-46
Author(s):  
DAVID PFRANG ◽  
MICHAEL ROTHGANG ◽  
DIERK SCHLEICHER

Abstract We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb {C}$ , do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a homotopy Hubbard tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a homotopy Hubbard tree that is unique up to homotopy, and that post-singularly finite exponential maps can be classified in terms of homotopy Hubbard trees, using a transcendental analogue of Thurston’s topological characterization theorem of rational maps.


2021 ◽  
Author(s):  
Tobias Rubel ◽  
Pramesh Singh ◽  
Anna Ritz

A major goal of molecular systems biology is to understand the coordinated function of genes or proteins in response to cellular signals and to understand these dynamics in the context of disease. Signaling pathway databases such as KEGG, NetPath, NCI-PID, and Panther describe the molecular interactions involved in different cellular responses. While the same pathway may be present in different databases, prior work has shown that the particular proteins and interactions differ across database annotations. However, to our knowledge no one has attempted to quantify their structural differences. It is important to characterize artifacts or other biases within pathway databases, which can provide a more informed interpretation for downstream analyses. In this work, we consider signaling pathways as graphs and we use topological measures to study their structure. We find that topological characterization using graphlets (small, connected subgraphs) distinguishes signaling pathways from appropriate null models of interaction networks. Next, we quantify topological similarity across pathway databases. Our analysis reveals that the pathways harbor database-specific characteristics implying that even though these databases describe the same pathways, they tend to be systematically different from one another. We show that pathway-specific topology can be uncovered after accounting for database-specific structure. This work present the first step towards elucidating common pathway structure beyond their specific database annotations.


2021 ◽  
Vol 140 (8) ◽  
Author(s):  
Micheal Arockiaraj ◽  
S. Prabhu ◽  
M. Arulperumjothi ◽  
S. Ruth Julie Kavitha ◽  
Krishnan Balasubramanian

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