The Tait First Conjecture for Alternating Weaving Diagrams.

Many entangled complex networks, like weaving frameworks, can be analyzed from a viewpoint of knot theory to better understand their topology. The number of crossings is in particular a suitable concept to study and classify such structures. In this paper, the Tait First Conjecture, which states that any reduced diagram of an alternating link has the minimal possible number of crossings, is extended to reduced alternating weaving diagrams on a higher genus surface, which lie on a surface of genus g defined either on the Euclidean plane or the hyperbolic plane . A weaving structure, also called weave, has many weaving diagrams on the plane associated to it, if the weave is constructed using a polygonal tessellation as a scaffold. The proof of the Tait First Conjecture for alternating weaves is inspired by the one for classical links from L.H. Kauffman, with an adaptation of the concepts of diagrams and invariants for weaving diagrams in higher genus surfaces, originally defined by S. Grishanov et al. in the case of genus one.

[1]  Guanyu Zhu,et al.  Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes , 2019, Quantum.

[2]  S. Hyde,et al.  A Theoretical Schema for Building Weavings of Nets via Colored Tilings of Two-Dimensional Spaces and Some Simple Polyhedral, Planar and Three-Periodic Examples , 2018, Israel Journal of Chemistry.

[3]  O. Yaghi,et al.  The geometry of periodic knots, polycatenanes and weaving from a chemical perspective: a library for reticular chemistry. , 2018, Chemical Society reviews.

[4]  Banglin Chen,et al.  Some equivalent two-dimensional weavings at the molecular scale in 2D and 3D metal–organic frameworks , 2016 .

[5]  S. Hyde,et al.  Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests. , 2015, Acta crystallographica. Section A, Foundations and advances.

[6]  A. Garciano,et al.  From Colorings to Weavings , 2014 .

[7]  Stephen T. Hyde,et al.  Periodic entanglement II: Weavings from hyperbolic line patterns , 2013 .

[8]  V. Robins,et al.  Periodic entanglement I: networks from hyperbolic reticulations , 2013 .

[9]  Benson Farb,et al.  A Primer on Mapping Class Groups (Pms-49) , 2011 .

[10]  Dan Lu,et al.  The architecture of Platonic polyhedral links , 2009 .

[11]  S. A. Grishanov,et al.  A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures , 2009 .

[12]  S. A. Grishanov,et al.  A Topological Study of Textile Structures. Part I: An Introduction to Topological Methods , 2009 .

[13]  Qiu Yuanyuan,et al.  Architecture of Platonic and Archimedean polyhedral links , 2008 .

[14]  S. A. Grishanov,et al.  KAUFFMAN-TYPE POLYNOMIAL INVARIANTS FOR DOUBLY PERIODIC STRUCTURES , 2007 .

[15]  Wen-yuan Qiu,et al.  Molecular design of Goldberg polyhedral links , 2005 .

[16]  Naoko Kamada.  Span of the Jones polynomial of an alternating virtual link , 2004, math/0412074.

[17]  On Knots , 1990, Acta Applicandae Mathematicae.

[18]  K. Murasugi.  Jones polynomials and classical conjectures in knot theory. II , 1987, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  Agnes Garciano,et al.  Construction of weavings in the plane. , 2018, Acta crystallographica. Section A, Foundations and advances.

[20]  A. Garciano,et al.  Equivalent weavings , 2017 .

[21]  Benson Farb,et al.  A primer on mapping class groups , 2013 .

[22]  Xiao-Sheng Cheng,et al.  Fabrication of a family of pyramidal links and their genus , 2010 .

[23]  Guang Hu,et al.  Topological transformation of dual polyhedral links , 2010 .

[24]  Ying-Qing Wu.  Jones polynomial and the crossing number of links , 1989 .

[25]  Morwen Thistlethwaite,et al.  Kauffman's polynomial and alternating links , 1988 .

[26]  Louis H. Kauffman,et al.  State Models and the Jones Polynomial , 1987 .

[27]  P. R. Lord.  1 – AN HISTORICAL INTRODUCTION TO WEAVING , 1982 .