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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.



A Survey of Hyperbolic Knot Theory

We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.
David Futer, Efstratia Kalfagianni, Jessica S. Purcell

Spanning Surfaces for Hyperbolic Knots in the 3-Sphere

We consider results and questions related to both the geometry and topology of surfaces that span hyperbolic knots, including embedded orientable and nonorientable surfaces as well as singular punctured surfaces.
Colin C. Adams

On the Construction of Knots and Links from Thompson’s Groups

We review recent developments in the theory of Thompson group representations related to knot theory.
Vaughan F. R. Jones

Virtual Knot Theory and Virtual Knot Cobordism

This paper is an introduction to virtual knot theory and virtual knot cobordism [37, 39]. Non-trivial examples of virtual slice knots are given and determinations of the four-ball genus of positive virtual knots are explained in relation to joint work with Dye and Kaestner [12]. We study the affine index polynomial [38], prove that it is a concordance invariant, show that it is invariant also under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. In particular we show how a mod-2 version of the affine index polynomial is a concordance invariant of flat virtual knots and links, and explore a number of examples in this domain.
Louis H. Kauffman

Knot Theory: From Fox 3-Colorings of Links to Yang–Baxter Homology and Khovanov Homology

This paper is an extended account of my “Introductory Plenary talk at Knots in Hellas 2016” conference. We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R. Llull (1232–1315), A. Kircher (1602–1680), Leibniz idea of Geometria Situs (1679), and J.B. Listing (student of Gauss) work of 1847. We spend some space on Ralph H. Fox (1913–1973) elementary introduction to diagram colorings (1956). In the second section we describe how Fox work was generalized to distributive colorings (racks and quandles) and eventually in the work of Jones and Turaev to link invariants via Yang–Baxter operators; here the importance of statistical mechanics to topology will be mentioned. Finally we describe recent developments which started with Mikhail Khovanov work on categorification of the Jones polynomial. By analogy to Khovanov homology we build homology of distributive structures (including homology of Fox colorings) and generalize it to homology of Yang–Baxter operators. We speculate, with supporting evidence, on co-cycle invariants of knots coming from Yang–Baxter homology. Here the work of Fenn–Rourke–Sanderson (geometric realization of pre-cubic sets of link diagrams) and Carter–Kamada–Saito (co-cycle invariants of links) will be discussed and expanded. No deep knowledge of Knot Theory, homological algebra, or statistical mechanics is assumed as we work from basic principles. Because of this, some topics will be only briefly described.
Józef H. Przytycki

Algebraic and Computational Aspects of Quandle 2-Cocycle Invariant

Quandle homology theories have been developed and cocycles have been used to construct invariants in state-sum form for knots using colorings of knot diagrams by quandles. Quandle 2-cocycles can be also used to define extensions as in the case of groups. There are relations among algebraic properties of quandles, their homology theories, and cocycle invariants; certain algebraic properties of quandles affect the values of the cocycle invariants, and identities satisfied by quandles induce subcomplexes of homology theory. Recent developments in these matters, as well as computational aspects of the invariant, are reviewed. Problems and conjectures pertinent to the subject are also listed.
W. Edwin Clark, Masahico Saito

A Survey of Quantum Enhancements

In this short survey article we collect the current state of the art in the nascent field of quantum enhancements, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes classical skein invariants and quandle and biquandle cocycle invariants as well as new invariants.
Sam Nelson

From Alternating to Quasi-Alternating Links

In this short survey, we introduce the class of quasi-alternating links and review some of their basic properties. In particular, we discuss the obstruction criteria for links to be quasi-alternating introduced recently in terms of quantum link invariants.
Nafaa Chbili

Hoste’s Conjecture and Roots of the Alexander Polynomial

The Alexander polynomial remains one of the most fundamental invariants of knots and links in 3-space. Its topological understanding has led a long time ago to the insight of what (Laurent) polynomials occur as Alexander polynomials of arbitrary knots. Ironically, the question to characterize the Alexander polynomials of alternating knots turns out to be far more difficult, even although in general alternating knots are much better understood. J. Hoste, based on computer verification, made the following conjecture about 15 years ago: If z is a complex root of the Alexander polynomial of an alternating knot, then \(\mathfrak {R}e\,z > -1\). We discuss some results toward this conjecture, about 2-bridge (rational) knots or links, 3-braid alternating links, and Montesinos knots.
Alexander Stoimenov

A Survey of Grid Diagrams and a Proof of Alexander’s Theorem

Grid diagrams are a representation of knot projections that are particularly useful as a format for algorithmic implementation on a computer. This paper gives an introduction to grid diagrams and demonstrates their programmable viability in an algorithmic proof of Alexander’s Theorem. Throughout, there are detailed comments on how to program a computer to encode the diagrams and algorithms.
Nancy C. Scherich

Extending the Classical Skein

We summarize the theory of a new skein invariant of classical links H[H] that generalizes the regular isotopy version of the Homflypt polynomial, H. The invariant H[H] is based on a procedure where we apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots and then we evaluate the resulting knots using the invariant H and inserting at the same time a new parameter. This procedure, remarkably, leads to a generalization of H but also to generalizations of other known skein invariants, such as the Kauffman polynomial. We discuss the different approaches to the link invariant H[H], the algebraic one related to its ambient isotopy equivalent invariant \(\Theta \), the skein-theoretic one and its reformulation into a summation of the generating invariant H on sublinks of a given link. We finally give examples illustrating the behaviour of the invariant H[H] and we discuss further research directions and possible application areas.
Louis H. Kauffman, Sofia Lambropoulou

From the Framisation of the Temperley–Lieb Algebra to the Jones Polynomial: An Algebraic Approach

We prove that the Framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley–Lieb algebras. We use this result to obtain a closed combinatorial formula for the invariants for classical links obtained from a Markov trace on the Framisation of the Temperley–Lieb algebra. For a given link L, this formula involves the Jones polynomials of all sublinks of L, as well as linking numbers.
Maria Chlouveraki

A Note on Link Invariants and the HOMFLY–PT Polynomial

We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY–PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the quantum \(\mathfrak {gl}_{m|n}\) invariants) as link invariants with values in the quantized oriented Brauer category.
Hoel Queffelec, Antonio Sartori

On the Geometry of Some Braid Group Representations

In this note we report on recent differential geometric constructions aimed at devising representations of braid groups in various contexts, together with some applications in different domains of mathematical physics. First, the classical Kohno construction for the 3- and 4-strand pure braid groups \(P_3\) and \(P_4\) is explicitly implemented by resorting to the Chen-Hain-Tavares nilpotent connections and to hyperlogarithmic calculus, yielding unipotent representations able to detect Brunnian and nested Brunnian phenomena. Physically motivated unitary representations of Riemann surface braid groups are then described, relying on Bellingeri’s presentation and on the geometry of Hermitian–Einstein holomorphic vector bundles on Jacobians, via representations of Weyl-Heisenberg groups.
Mauro Spera

Towards a Version of Markov’s Theorem for Ribbon Torus-Links in

In classical knot theory, Markov’s theorem gives a way of describing all braids with isotopic closures as links in \(\mathbb {R}^3\). We present a version of Markov’s theorem for extended loop braids with closure in \(B^3 \times S^1\), as a first step towards a Markov’s theorem for extended loop braids and ribbon torus-links in \(\mathbb {R}^4\).
Celeste Damiani

An Alternative Basis for the Kauffman Bracket Skein Module of the Solid Torus via Braids

In this paper we give an alternative basis, \(\mathscr {B}_\mathrm{ST}\), for the Kauffman bracket skein module of the solid torus, \(\mathrm{KBSM}\left( \mathrm{ST}\right) \). The basis \(\mathscr {B}_\mathrm{ST}\) is obtained with the use of the Tempereley–Lieb algebra of type B and it is appropriate for computing the Kauffman bracket skein module of the lens spaces L(pq) via braids.
Ioannis Diamantis

Knot Invariants in Lens Spaces

In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent by mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.
Boštjan Gabrovšek, Eva Horvat

Identity Theorem for Pro-p-groups

The concept of schematization consists in replacing simplicial groups by simplicial affine group schemes. In the case when the coefficient field has zero characteristic, there is a prominent theory of simplicial prounipotent groups, the origins of which lead to the rational homotopy theory of D. Quillen. It turns out that schematization reveals the profound properties of \(\mathbb {F}_p\)-prounipotent groups, especially in connection with prounipotent groups in zero characteristic and in the study of quasirationality. In this paper, using results on representations and cohomology of prounipotent groups in characteristic 0, we prove an analogue of Lyndon Identity theorem for one-relator pro-p-groups (question posed by J.P. Serre) and demonstrate the application to one more problem of J.-P. Serre concerning one-relator pro-p-groups of cohomological dimension 2. Schematic approach makes it possible to consider the problems of pro-p-groups theory through the prism of Tannaka duality, concentrating on the category of representations. In particular we attach special importance to the existence of identities in free pro-p-groups (“conjurings”).
Andrey M. Mikhovich

A Survey on Knotoids, Braidoids and Their Applications

This paper is a survey on the theory of knotoids and braidoids. Knotoids are open ended knot diagrams in surfaces and braidoids are geometric objects analogous to classical braids, forming a counterpart theory to the theory of knotoids in the plane. We survey through the fundamental notions and existing works on these objects as well as their applications in the study of proteins.
Neslihan Gügümcü, Louis H. Kauffman, Sofia Lambropoulou

Regulation of DNA Topology by Topoisomerases: Mathematics at the Molecular Level

Although the genetic information is encoded in a one-dimensional array of nucleic acid bases, three-dimensional relationships within DNA play a major role in how this information is accessed and utilized by living organisms. Because of the intertwined nature of the DNA two-braid and its extreme length and compaction in the cell, some of the most important three-dimensional relationships in DNA are topological in nature. Topological linkages within the two-braid and between different DNA segments can be described in simple mathematical terms that account for both the twist and the writhe in the double helix. Topoisomerases are ubiquitous enzymes that regulate the topological state of the genetic material by altering either twist or writhe. To do so, these enzymes transiently open the topological system by breaking one or both strands of the two-braid. This article will review the mathematics of DNA topology, describe the different classes of topoisomerases, and discuss the mechanistic basis for their actions in both biological and mathematical terms. Finally, it will discuss how topoisomerases recognize the topological states of their DNA substrates and products and how some of these enzymes distinguish supercoil handedness during catalysis and DNA cleavage. These latter characteristics make topoisomerases well suited for their individual physiological tasks and impact their roles as targets of important anticancer and antibacterial drugs.
Rachel E. Ashley, Neil Osheroff

Topological Entanglement and Its Relation to Polymer Material Properties

In this manuscript we review recent results that show how measures of topological entanglement can be used to provide information relevant to dynamics and mechanics of polymers. We use Molecular Dynamics simulations of coarse-grained models of polymer melts and solutions of linear chains in different settings. We apply the writhe to give estimates of the entanglement length and to study the disentanglement of polymer melts in an elongational flow. Our results also show that our topological measures correlate with viscoelastic properties of the material.
Eleni Panagiotou

Topological Surgery in the Small and in the Large

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.
Stathis Antoniou, Louis H. Kauffman, Sofia Lambropoulou


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