Open Questions: Knot Theory

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O time! Thou must untangle this, not I;
It is too hard a knot for me t'untie.
William Shakespeare, Twelfth Night

Introduction

Recommended references: Web sites

Recommended references: Magazine/journal articles

Recommended references: Books

Introduction

Recommended references: Web sites

Site indexes

Open Directory Project: Knot Theory: Categorized and annotated knot theory links. A version of this list is at Google, with entries sorted in "page rank" order.
Knots on the Web: An extensive page of links to anything and everything about knots. Includes knot art and how to tie specific knots, in addition to the mathematical theory.
The Net Advance of Physics: Knot Theory: An index of tutorial and research articles located at the physics preprint archive. Emphasis is on applications in physics.
Knot Theory: Short list, provided by Mark Brittenham.

Sites with general resources

The Knot Theory Home Page: Good tutorial information on knots in general, knot invariants, and applications of knot theory.
The KnotPlot Site: KnotPlot is a software package for rendering images of knots from a large database and knots that can be constructed within the package. The site contains images and knot theory information. The software was developed by Robert Scharein and may be downloaded from the site.

Surveys, overviews, tutorials

Knot theory: Article from Wikipedia.
Knots and their Polynomials: Good overview of knot theory and topological invariants of knots, the Jones polynomial in particular.
Unknotting Knot Theory: Excellent article by Julie Rehmeyer that explaines a lot about knot invariants, including some of the latest developments, such as Khovanov and Floer homology theories.
The Knot Theory MA3F2 page: A course in knot theory, including supplementary material, exams and solutions, by Brian Sanderson.
Invariants of knots and 3-manifolds (Kyoto 2001): Collection of online technical papers from a conference, part of the Geometry and Topology Monographs series.

Recommended references: Magazine/journal articles

The Combinatorial Revolution in Knot Theory
Sam Nelson
Notices of the American Mathematical Society, December 2011: Much as the concept of "numbers" has evolved over time from its original meaning of cardinalities of finite sets to include ratios, equivalence classes of rational Cauchy sequences, roots of polynomials, and more, the classical concept of "knots" has recently undergone its own evolutionary generalization.
Knotty Calculations
Erica Klarreich
Science News, February 22, 2003, pp. 124-126: Nature can be thought of as a computer which can solve enormously complicated equations, such as those governing the motions of all planets and asteroids in the solar system. Certain kinds of systems known as "fractional quantum Hall fluids" may be able to solve equations in the mathematical theory of braids, which could make possible the computation of the knot invariants known as "Jones polynomials", and consequently be able to solve manhy other hard problems as well.
Knot Possible
Ivars Peterson
Science News, December 8, 2001, pp. 360-361: There has been recent progress in developing practical procedures for distinguishing knotted curves from unknotted ones.
[References]
How Hard Is It to Untie a Knot?
William Menasco; Lee Rudolph
American Scientist, January-February 1995, pp. 38-49: A question about knots known as the Bennequin conjecture has recently been resolved. The proof also answers a question of Milnor related to issues in physics and exotic spheres. There are applications of these resultes to the unknotting of DNA.
New Points of View in Knot Theory
Joan S. Birman
Bulletin of the AMS, April 1993, pp. 253-287: A survey of developments in know theory following the 1984 discovery of the Jones polynomials. There are three themes: (1) Trying to understand the topological meaning of the new invariants, (2) The central role of braid theory in the subject, (3) Unifying principles provided by representations of simple Lie algebras and their universal enveloping algebras.
Review of The geometry and physics of knots by Michael Atiyah
Raoul Bott
Bulletin of the AMS, January 1992, pp. 182-188: Within the last decade several interesting new manifold invariants have been discovered, such as Donaldson's 4-manifold invariants and the knot polynomials of Vaughan Jones. These developments have been stimulated by ideas from physics. The subject is now a hybrid of statistical mechanics, algebraic topology, and Lie group representation theory.

Recommended references: Books

Alexei Sossinski -- Knots: Mathematics with a Twist
Harvard University Press, 2002: Sossinski is an expert in knot theory. This brief book introduces the essential mathematical ideas without a lot of excess baggage. The main focus is on the matematical description of knots, and especially the various invariants. There's a short concluding chapter on the role of knots in modern physics.
W. B. Raymond Lickorish -- An Introduction to Knot Theory
Springer-Verlag, 1997: This is a graduate-level mathematics text and assumes some exposure to ideas of algebraic topology. So it's not for the general reader, but if the level of sophistication isn't a deterrent, you can find here a rigorous introduction to the theory, as well as recent theoretical developments.
Colin C. Adams -- The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
W. H. Freeman and Company, 1994: Like Sossinksi's book, there's real mathematics inside, and a lot more of it, since the book is almost three times as long. Topics include notations for knots, simple invariants, polynomial invariants, knots and graphs, knots and topology, and knots in higher dimensions. There's also a chapter on knots in biology, chemistry, and physics.

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