Knots, Low-Dimensional Topology and Applications: Knots in Hellas, International Olympic Academy, Greece, July 2016

Description

This proceedings volume presents a diverse collection of high-quality, state-of-the-art research and survey articles written by top experts in low-dimensional topology and its applications. The focal topics include the wide range of historical and contemporary invariants of knots and links and related topics such as three- and four-dimensional manifolds, braids, virtual knot theory, quantum invariants, braids, skein modules and knot algebras, link homology, quandles and their homology; hyperbolic knots and geometric structures of three-dimensional manifolds; the mechanism of topological surgery in physical processes, knots in Nature in the sense of physical knots with applications to polymers, DNA enzyme mechanisms, and protein structure and function. The contents is based on contributions presented at the International Conference on Knots, Low-Dimensional Topology and Applications – Knots in Hellas 2016, which was held at the International Olympic Academy in Greece in July 2016. The goal of the international conference was to promote the exchange of methods and ideas across disciplines and generations, from graduate students to senior researchers, and to explore fundamental research problems in the broad fields of knot theory and low-dimensional topology. This book will benefit all researchers who wish to take their research in new directions, to learn about new tools and methods, and to discover relevant and recent literature for future study. Front Matter ....Pages i-xii A Survey of Hyperbolic Knot Theory (David Futer, Efstratia Kalfagianni, Jessica S. Purcell)....Pages 1-30 Spanning Surfaces for Hyperbolic Knots in the 3-Sphere (Colin C. Adams)....Pages 31-42 On the Construction of Knots and Links from Thompson’s Groups (Vaughan F. R. Jones)....Pages 43-66 Virtual Knot Theory and Virtual Knot Cobordism (Louis H. Kauffman)....Pages 67-114 Knot Theory: From Fox 3-Colorings of Links to Yang–Baxter Homology and Khovanov Homology (Józef H. Przytycki)....Pages 115-145 Algebraic and Computational Aspects of Quandle 2-Cocycle Invariant (W. Edwin Clark, Masahico Saito)....Pages 147-162 A Survey of Quantum Enhancements (Sam Nelson)....Pages 163-178 From Alternating to Quasi-Alternating Links (Nafaa Chbili)....Pages 179-189 Hoste’s Conjecture and Roots of the Alexander Polynomial (Alexander Stoimenov)....Pages 191-206 A Survey of Grid Diagrams and a Proof of Alexander’s Theorem (Nancy C. Scherich)....Pages 207-224 Extending the Classical Skein (Louis H. Kauffman, Sofia Lambropoulou)....Pages 225-245 From the Framisation of the Temperley–Lieb Algebra to the Jones Polynomial: An Algebraic Approach (Maria Chlouveraki)....Pages 247-276 A Note on \(\mathfrak {gl}_{m|n}\) Link Invariants and the HOMFLY–PT Polynomial (Hoel Queffelec, Antonio Sartori)....Pages 277-286 On the Geometry of Some Braid Group Representations (Mauro Spera)....Pages 287-308 Towards a Version of Markov’s Theorem for Ribbon Torus-Links in \(\mathbb {R}^4\) (Celeste Damiani)....Pages 309-328 An Alternative Basis for the Kauffman Bracket Skein Module of the Solid Torus via Braids (Ioannis Diamantis)....Pages 329-345 Knot Invariants in Lens Spaces (Boštjan Gabrovšek, Eva Horvat)....Pages 347-361 Identity Theorem for Pro-p-groups (Andrey M. Mikhovich)....Pages 363-387 A Survey on Knotoids, Braidoids and Their Applications (Neslihan Gügümcü, Louis H. Kauffman, Sofia Lambropoulou)....Pages 389-409 Regulation of DNA Topology by Topoisomerases: Mathematics at the Molecular Level (Rachel E. Ashley, Neil Osheroff)....Pages 411-433 Topological Entanglement and Its Relation to Polymer Material Properties (Eleni Panagiotou)....Pages 435-447 Topological Surgery in the Small and in the Large (Stathis Antoniou, Louis H. Kauffman, Sofia Lambropoulou)....Pages 449-456 Back Matter ....Pages 457-476