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1
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Properties of Topological Networks of Flexible Polygonal Chains

100%
Arsuaga J., Diao Y., Klingbeil M., Rodriguez V.
Molecular Based Mathematical Biology
|
2014
|
tom 2
|
nr 1
EN
Trypanosomatida parasites, such as Trypanosoma and Leishmania, are the cause of deadly diseases in many third world countries. The three dimensional structure of their mitochondrial DNA, known as kinetoplast DNA (kDNA), is unique since it is organized into several thousands of minicircles that are topologically linked. How and why the minicircles form such a network have remained unanswered questions. In our previous work we have presented a model of network formation that hypothesizes that the network is solely driven by the confinement of minicircles. Our model shows that upon confinement a percolation network forms. This network grows into a space filling network, called saturation network, upon further confinement of minicircles. Our model also shows, in agreement with experimental data, that the mean valence of the network (that is, the average number of minicircles topologically linked to any minicircle in the network) grows linearly with minicircle density. In our previous studies however we disregarded DNA flexibility and used rigid minicircles to model DNA, here we address this limitation by allowing minicircles to be flexible. Our numerical results show that the topological characteristics that describe the growth and topology of the minicircle networks have similar values to those observed in the case of rigid minicircles suggesting that these properties are robust and therefore a potentially adequate description of the networks observed in Trypanosomatid parasites.
2
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Notes on tiled incompressible tori

100%
Plachta L.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2200-2210
EN
Let Θ denote the class of essential tori in a closed braid complement which admit a standard tiling in the sense of Birman and Menasco [Birman J.S., Menasco W.W., Special positions for essential tori in link complements, Topology, 1994, 33(3), 525–556]. Moreover, let R denote the class of thin tiled tori in the sense of Ng [Ng K.Y., Essential tori in link complements, J. Knot Theory Ramifications, 1998, 7(2), 205–216]. We define the subclass B ⊂ Θ of typical tiled tori and show that R ⊂ B. We also describe a method allowing to construct new examples of tiled essential tori T which are outside the class B in the strong sense. In [Kazantsev A., Essential tori in link complements: detecting the satellite structure by monotonic simplification, preprint available at http://arxiv.org/abs/1005.5263], Kazantsev showed that the inclusion R ⊂ Θ is proper by giving the corresponding example of a nonthin tiled torus T. It turns out this torus T is inside the class B. We show that the inclusion B ⊂ Θ is proper. It follows that the tori from the class B do not provide the complete geometric description of the class Θ. The main results of the paper are Theorems 2.1 and 2.2 which give a constructive procedure for obtaining examples of nontypical tiled essential tori.
3
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A study about the Tutte polynomials of benzenoid chains

76%
Sahin A.
Topological Algebra and its Applications
|
2017
|
tom 5
|
nr 1
28-32
EN
The Tutte polynomials for signed graphs were introduced by Kauffman. In 2012, Fath-Tabar, Gholam-Rezaeı and Ashrafı presented a formula for computing Tutte polynomial of a benzenoid chain. From this point on, we have also calculated the Tutte polynomials of signed graphs of benzenoid chains in this study.
4
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The classification of partially symmetric 3-braid links

64%
Stoimenov A.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
We classify 3-braid links which are amphicheiral as unoriented links, including a new proof of Birman- Menasco’s result for the (orientedly) amphicheiral 3-braid links. Then we classify the partially invertible 3-braid links.
5
Content available remote

Discrete thickness

64%
Scholtes S.
Molecular Based Mathematical Biology
|
2014
|
tom 2
|
nr 1
EN
We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are deffned on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the discrete ropelength for n → ∞, regarding the topology induced by the Sobolev norm ‖ · ‖ W1,∞(S1,ℝd). This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the smooth energy.Moreover,we show that the unique absolute minimizer of inverse discrete thickness is the regular n-gon.
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