リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「Construction and classification of topological weaves」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。
リケラボ

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

Construction and classification of topological weaves

Mahmoudi Sonia 東北大学

2022.09.26

概要

Weaves are complex entangled objects that mainly differ from general links because they contain no closed components. Although weaves have been investigated for so many years, we still do not have a universal study to describe them. Several interesting attempts have been made to approach them from a mathematical point of view and this thesis contributes to it by introducing a new way to define, construct, and classify topological weaves.

In Chapter 2, based on our paper [21], we state a new definition of weaves as the lift to the thickened Euclidean plane X 3 of a quadrivalent graph Γ in E 2 made of colored straight lines, such that each vertex is specified by an over or under information given by a set of crossing sequences Σ.

Definition 0.1. [21] We call untwisted weave the lift to X 3 of a pair (Γ, Σ) of E 2 . Moreover, two threads are said to be in the same set of threads, if they are the lift of straight lines belonging to the same color group.

More complex weaves, called twisted weaves, can be defined from these untwisted ones by introducing twists between neighboring threads of the same color, via some local surgeries, called ±k-moves in knot theory.

Definition 0.2. [21] A twisted weave is the lift to X 3 of a pair (Γ, Σ) embedded on E 2 admitting at least a twisted region. Moreover, if two threads twist their total number of twists is even and they cannot twist with other threads.

We also define diagrammatic representations to study their properties.

Definition 0.3. [21] The planar projection W0 of a weave W by π : X 3 → E 2 , (x, y, z) 7→ (x, y, 0) is called a regular projection. When all its vertices are specified by an over or under information, it is called a weaving diagram DW0 . And if DW0 is periodic, then any generating cell is called a weaving motif.

In Chapter 3, we introduce a systematic way to construct weaving motifs [22], based on the concept of polyhedral link defined by W.Y. Qiu et al. [47]. The strategy consists in covering the edges and vertices of a generating cell UT of a periodic tiling T of E 2 by crossed strands with respect to the polygonal link methods, denoted by (Λ, L). However, we have noticed that these methods can generate closed curves and have developed a way to predict the construction of weaving motifs using algebraic and combinatorial arguments.

Theorem 0.4. [22] (Construction of Weaving Motifs) A weaving motif is created from a pair UT ,(Λ, L) if and only if all the characteristic loops in UT are nontrivial polygonal chains with at least two non-equivalent ones

In Chapter 4, we take a first step towards the classification of alternating weaving motifs [48], via the extension of two famous theorems of knot theory. By alternating, we mean that the crossings alternate cyclically between under and over. To achieve this goal, we generalize the concept of reduced diagram to periodic weaves and extend the Kauffman bracket polynomial [39], defined for weaving motifs on a torus in [25], to higher genus surfaces Σg

Theorem 0.5. [48] (Tait’s First and Second Conjectures for Weaves) The crossing number and the writhe of a Σg-reduced alternating minimal diagram of its periodic alternating weaving diagram are weaving invariants

Chapter 5 ends this manuscript with a strong result in terms of classification of the class of doubly periodic untwisted (p, q)-weaves, based on our paper [21]. First, we construct a new weaving invariant defined as a set of crossing matrices, whose elements are symbols ±1 characterizing the organization of crossings in a generating cell. Then, we prove our main theorem.

Theorem 0.6. [21] (Equivalence Classes of Doubly Periodic Untwisted (p, q)-Weaves) Let W1 and W2 be two doubly periodic untwisted (p, q)-weaves with N ≥ 2 sets of threads, such that their corresponding regular projections are equivalent, up to isotopy of E 2 , and with the same set of crossing sequences. Let DW1 and DW2 be two weaving motifs of same area of W1 and W2, respectively. Then, DW1 and DW2 are equivalent if and only if their crossing matrices are pairwise equivalent.

Then, in a logic of classification in terms of crossings, we state a solution to the open problem of finding the crossing number of weaving motifs for doubly periodic untwisted (p, q)-weaves. The idea is to use combinatorial arguments on the torus to obtain a formula which depends on (Γ, Σ)

論文の公開元へ

関連論文

関連論文をもっと見る

参考文献

[1] C. Adams. The knot book: an elementary introduction to the mathematical theory of knots. (American Mathematical Society, Providence, 2004).

[2] C. Adams, C. Albors-Riera, B. Haddock, Z. Li, D. Nishida, B. Reinoso, L. Wang. Hyperbolicity of links in thickened surfaces. Topology and Its Applications. 256 (2019), 262–278.

[3] C. Adams, A. Calderon, N. Mayer. Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links. Topology and Its Applications. 271 (2020),107045.

[4] C. Adams, T. Fleming, M. Levin, and A. Turner. Crossing number of alternating knots in S x I. Pac. J. Math. 203 (2002), 1–22.

[5] H.U. Boden and H. Karimi. The Jones-Krushkal polynomial and minimal diagrams of surface links. arXiv:1908.06453v1. (2019).

[6] M. Bright, V. Kurlin. Encoding and topological computation on textile structures. Computers and Graphics. 90 (2020) 51–61.

[7] G. Brinkmann, M. Deza. Lists of face-regular polyhedra. J. Chem. Inf. Comput. Sci. 40 (2000), 530–541.

[8] T. Castle, M.E. Evans, S.T. Hyde. Entanglement of embedded graphs. Prog. Theor. Phys. Suppl. 191 (2011), 235–244.

[9] A. Champanerkar, I. Kofman, J.S. Purcell. Volume bounds for weaving knots. Algebr. Geom. Topol. 16 (2016), 3301–3323.

[10] A. Champanerkar, I. Kofman, J.S. Purcell. Geometrically and diagrammatically maximal knots. J. London Math. Soc. 94 (2016), 883–908.

[11] J.H. Conway, H. Burgiel. C. Goodman-Strauss. The symmetries of things. (A K Peters/CRC Press, 2016).

[12] P.R. Cromwell. Knots and Links, 1st ed. (Cambridge University Press, 2004).

[13] S. De Toffoli, V. Giardino. Forms and Roles of Diagrams in Knot Theory. Erkenn. 79 (2014), 829–842.

[14] A.-M. D´ecaillot. G´eom´etrie des tissus. Mosa¨ıques. Echiquiers. Math´ematiques ´ curieuses et utiles. Revue d’hist. Math. 8 (2002), 145–206.

[15] T. Endo, T. Itoh, K. Taniyama. A graph-theoretic approach to a partial order of knots and links. Topology and Its Applications. 157 (2010), 1002–1010.

[16] M.E. Evans, V. Robins and S.T. Hyde. Periodic entanglement I: networks from hyperbolic reticulations. Acta Cryst. A69 (2013), 241–261.

[17] M.E. Evans, V. Robins and S.T. Hyde. Periodic entanglement II: weavings from hyperbolic line patterns. Acta Cryst. A69 (1984), 262–275.

[18] M.E. Evans and S.T. Hyde. Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests. Acta Cryst. A71 (2015),599–611.

[19] B. Farb and D. Margalit. A primer on mapping class groups. (Princeton University Press, Princeton, 2012).

[20] T. Fleming. Strict Minimality of Alternating Knots in S × I. J. Knot Theory Its Ramif. 12 (2003),445–462.

[21] M. Fukuda, M. Kotani and S. Mahmoudi. Classification of doubly periodic untwisted (p, q)-weaves by their crossing number. arXiv:2202.01755 (2022).

[22] M. Fukuda, M. Kotani and S. Mahmoudi. Construction of weaving and polycatenanes motifs from periodic tilings of the plane. arXiv:2206.12168 (2022).

[23] S. Grishanov, V. Meshkov and A. Omelchenko. A topological study of textile structures. Part I: an introduction to topological methods. Text. Res. J. 79 (2009),702–713.

[24] S. Grishanov, V. Meshkov and A. Omelchenko. A topological study of textile structures. Part II: topological invariants in application to textile structures. Text. Res. J. 79 (2009),822–836.

[25] S.A. Grishanov, V.R. Meshkov, A.V. Omel’Chenko. Kauffman-type polynomial invariants for doubly periodic structures. J. Knot Theory Ramifications. 16 (2007), 779–788.

[26] S.A. Grishanov, V.R. Meshkov, V.A. Vassiliev. Recognizing textile structures by finite type knot invariants. J. Knot Theory Ramifications. 18 (2009), 209–235.

[27] S.A. Grishanov, V.A. Vassiliev. Invariants of links in 3-manifolds and splitting problem of textile structures. J. Knot Theory Ramifications. 20 (2011), 345–370.

[28] B. Gr¨unbaum and G.C. Shephard. A catalogue of isonemal fabrics. in Discrete Geometry and Convexity (J. E. Goodman et al., eds); Ann. New York Acad. Sci. 440 (1985), 279–298.

[29] B. Gr¨unbaum and G.C. Shephard. An extension to the catalogue of isonemal fabrics. Discrete Math. 60 (1986), 155–192.

[30] B. Gr¨unbaum and G.C. Shephard. Isonemal fabrics. The American Math. Monthly. 95 (1988), 5–30.

[31] B. Gr¨unbaum and G.C. Shephard. Tilings and Patterns (second edition). (Dover Publications, New York, 2016).

[32] B. Gr¨unbaum and G.C. Shephard. Satins and Twills: an introduction to the geometry of fabrics. Math. Mag. 53 (1980), 139–161.

[33] G. Hu, X.-D. Zhai, D. Lu and W.-Y. Qiu. The architecture of Platonic polyhedral links. J. Math. Chem. 46 (2009), 592–603.

[34] S.T. Hyde, B. Chen and M. O’Keeffe. Some equivalent two-dimensional weavings at the molecular scale in 2D and 3D metal–organic frameworks. CrystEngComm. 18 (2016), 7607–7613.

[35] S.T. Hyde, M.E. Evans. Symmetric tangled Platonic polyhedra. Proc. Natl. Acad. Sci. U.S.A. 119 (2022), e2110345118.

[36] S. Ikeda, M. Kotani. A New Direction in Mathematics for Materials Science. (Springer Japan, Tokyo, 2015).

[37] N. Kamada. Span of the Jones polynomial of an alternating virtual link. Algebr. Geom. Topol. 4 (2004), 1083–1101.

[38] L. H. Kauffman. Introduction to virtual knot theory. J. Knot Theory Ramifications. 21 (2012), 1240007.

[39] L. H. Kauffman. State models and the jones polynomial. Topology. 26 (1987), 395– 407.

[40] A. Kawauchi. Complexities of a knitting pattern. Reactive and Functional Polymers. 131 (2018) 230–236.

[41] K. Kobata, T. Tanaka. A circular embedding of a graph in Euclidean 3-space. Topology and Its Applications. 157 (2010), 213–219.

[42] S. Lambropoulou. Diagrammatic Representations of Knots and Links as closed braids. ArXiv:1811.11701. (2018).

[43] A. Lavasani, G. Zhu and M. Barkeshli. Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes. Quantum. 3 (2019), 180.

[44] W.B.R. Lickorish. An introduction to knot theory. (Springer New York, New York, 1997).

[45] Y. Liu, M. O’Keeffe, M.M.J. Treacy and O.M. Yaghi. The geometry of periodic knots, polycatenanes and weaving from a chemical perspective: a library for reticular chemistry. Chem. Soc. Rev. 47 (2018), 4642–4664.

[46] P.R. Lord and M.H. Mohamed. An historical introduction to weaving. in: Weaving, (Elsevier, 1982), 1–16.

[47] D. Lu, G. Hu, Y.-Y. Qiu and W.-Y. Qiu. Topological transformation of dual polyhedral links. MATCH Commun. Math. Comput. Chem. 63 (2010), 67–78.

[48] S. Mahmoudi. The Tait First and Second Conjecture for Alternating Weaving Diagrams. arXiv:2009.13896 (2021).

[49] W.W. Menasco, M.B. Thistlethwaite. The Classification of Alternating Links. The Annals of Mathematics 138 (1993), 113.

[50] W.W. Menasco, M.B. Thistlethwaite. Handbook of knot theory. (Elsevier, Amsterdam Boston, 2005).

[51] W.W. Menasco, M.B. Thistlethwaite. The Tait flyping conjecture. Bull. Amer. Math. Soc. 25 (1991), 403–413.

[52] E.D. Miro, A. Garciano and A. Zambrano. From colorings to weavings. Acta Phys. Pol. A. 126 (2014), 560–563.

[53] E.D. Miro, A. Zambrano and A. Garciano. Construction of weavings in the plane. Acta Cryst. A74 (2018), 25–35.

[54] H.R. Morton, S.A. Grishanov. Doubly periodic textile structures. J. Knot Theory Ramifications. 18 (2009), 1597–1622.

[55] K. Murasugi. Jones polynomials and classical conjectures in knot theory. Topology. 26 (1987), 187–194.

[56] K. Murasugi. Jones polynomials and classical conjectures in knot theory. II. Math. Proc. Camb. Philos. Soc. 102 (1987) 317–318.

[57] K. Murasugi. Knot theory and its applications (Birkh¨auser, Boston 2008).

[58] V.M. Ngo, S. Helmer, N.-A. Le-Khac, M.-T. Kechadi. Structural textile pattern recognition and processing based on hypergraphs. Inf Retrieval J. 24 (2021) 137–173.

[59] E. Panagiotou, K.C. Millett, S. Lambropoulou, The linking number and the writhe of uniform random walks and polygons in confined spaces. J. Phys. A: Math. Theor. 43 (2010), 045208.

[60] W.-Y. Qiu, X.-D. Zhai and Y.-Y. Qiu. Architecture of Platonic and Archimedean polyhedral links. J. Math. Chem. 51 (2008), 13–18.

[61] A. Schrijver. Tait’s flyping conjecture for well-connected links. J. of Combinatorial Theory, Series B. 58 (1993) 65–146.

[62] S. Sekhri. Textbook of Fabric Science : Fundamentals to Finishing (third edition). (PHI Learning Pvt. Ltd., 2019).

[63] K. Shimokawa, K. Ishihara, Y. Tezuka. Topology of polymers. (Springer Nature, Tokyo, 2019).

[64] R. Stong, The Jones polynomial of parallels and applications to crossing number, Pac. J. Math. 164 (1994) 383–395.

[65] P.G. Tait. On Knots I, II, III. in: Sci. Pap. Vol I, (Cambridge University Press, London, 1898), 273–347.

[66] M.B. Thistlethwaite. Kauffman’s polynomial and alternating links. Topology. 27 (1988), 311–318.

[67] B. Thompson and S.T. Hyde. A theoretical schema for building weavings of nets via colored tilings of two-dimensional spaces and some simple polyhedral, planar and three-periodic examples. Isr. J. Chem. 58 (2018), 1144–1156.

[68] P. Wadekar, C. Amanatides, L. Kapllani, G. Dion, R. Kamien, D.E. Breen. Geometric modeling of complex knitting stitches using a bicontinuous surface and its offsets. Comput. Aided Geometric Design. 89 (2021) 102024.

[69] P. Wadekar, P. Goel, C. Amanatides, G. Dion, R.D. Kamien, D.E. Breen, Geometric modeling of knitted fabrics using helicoid scaffolds, J. of Engineered Fibers and Fabrics. 15 (2020) 155892502091387.

[70] N. Wakayama, K. Shimokawa. On the classification of polyhedral links. Submitted. (2022).

[71] Y. Wu. Jones polynomial and the crossing number of links. in: Differ. Geom. Topol., (Springer Berlin Heidelberg, Berlin, 1989), 286–288.

[72] A. Zambrano, E.D. Miro and A. Garciano. Equivalent weavings. J. Phys.: Conf. Ser. 893 (2017), 012007.

参考文献をもっと見る