Jozef H. Przytycki
We present in this chapter (Chapter II) the history of ideas which lead up to the development of modern knot theory. We are more detailed when pre-XX century history is reported. With more recent times we are more selective, stressing developments related to Jones type invariants of links. In the Appendix, A.Przybyszewska translation of Preface to P.Heegaard Dissertation (1898) is given.
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This is a chapter that is to appear in the "Handbook of Knot Theory", edited by William W. Menasco and Morwen B. Thistlethwaite.
The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of such manifolds and knots. The first chapter is purely algebraic. The rest of the book may be divided into three parts: general results on homotopy and surgery (Chapters 2-6), geometries and geometric decompositions (Chapters 7-13), and 2-knots (Chapters 14-18). In many cases the Euler characteristic, fundamental group and Stiefel-Whitney classes together for…
Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics
本文是一篇指导研究资格论文，概述了表示论中Hecke代数与中心化子的理论，包括Schur-Weyl对偶、Yang-Baxter方程解的构造、李代数对角中心化子及其在纽结不变量与物理中的应用。
This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum. There is a folklore result in physics saying that the Feynman integration of an exponential is itself an exponential. In chapter 2 we state and prove an exact formulation of this statement in the language which is used in the theory of finit…
This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox 3-coloring. We show also a relation between 3-colorings and the Jones polynomial. In the second part we develop the general theory of Fox colorings and show how to associate a symplectic structure to a tangle boundary so that tangles becomes Lagrangians (a proof of this result has not been published before). Chapter VI of the book "KNOTS: From combinatorics of kn…
This chapter from the upcoming Handbook of Knot Theory (eds. Menasco and Thistlethwaite) shows how to construct hyperbolic structures on link complements and perform hyperbolic Dehn filling. Along with a new elementary exposition of the standard ideas from Thurston's work, the article includes never-before-published explanations of SnapPea's algorithms for triangulating a link complement efficiently and for converging quickly to the hyperbolic structure while avoiding singularities in the parameter space.
The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an independent, geometrically pleasing and consistent discrete model that behaves similarly to the original model. We will focus on Möbius energy, integral Menger curvature and thickness.
In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field theory framework of Wehrheim and Woodward to define a new homological knot invariant $\mathcal{S}(K)$. We then use a result of Ivan Smith to show that when $K$ is a (1,1) knot in $S^3$ (a set of knots which includes torus knots, for example), the rank of $\mathcal{S}(K)\otimes \mathbb{C}$ agrees with the rank of knot Floer homology, $\widehat{HFK}(K)\otimes \mat…
A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropelength and distortion of a knot.
In this chapter (Chapter V) we present several results which demonstrate a close connection and useful exchange of ideas between graph theory and knot theory. These disciplines were shown to be related from the time of Tait (if not Listing) but the great flow of ideas started only after Jones discoveries. The first deep relation in this new trend was demonstrated by Morwen Thistlethwaite and we describe several results by him in this Chapter. We also presentresults from two (unpublished) preprints [P-P-0,P-18].In particular, in Section 2, we sketch two generalizations of the Tutte polynomial o…
We describe in this chapter (Chapter IX) the idea of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, our basic building blocks are considered up to ambient isotopy (not homotopy or homology). For example, one should start from knots in 3-manifolds, surfaces in 4-manifolds, etc. However our theory is, until now, developed only in the case of links in 3-manifolds, with only a glance towards 4-manifolds. The main object of the theory is a skein module and we devote this chapter mostly to the description of skein modules in 3-dimensio…
The survey we are presenting is over 22 years old but it has still some ideas which where never published (except in Polish). This survey is the base of the third Chapter of my book: KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots, which is still in preparation (but compare http://arxiv.org/pdf/math.GT/0512630). The purpose of this survey is to present a new combinatorial method of constructing invariants of isotopy classes of tame links. The period of time between the spring of 1984 and the summer of 1985 was full of discoveries which revolutionized the kno…
The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov f…
This is a list of open problems on invariants of knots and 3-manifolds with expositions of their history, background, significance, or importance. This list was made by editing open problems given in problem sessions in the workshop and seminars on `Invariants of Knots and 3-Manifolds' held at Kyoto in 2001.
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