Category theory for knot theory mathematics stack exchange. Knot theory is of central importance in pure and applied mathematics, as it stands at a crossroads of topology, combinatorics, algebra, mathematical physics and biochemistry. Trefoil knot fi g ure 4 c lo sing b ra ids to fo rm k no ts a nd link s. An introduction to knot theory graduate texts in mathematics. Elementary constructions of homfly and kau man polynomials l.
The big problem in knot theory is finding out whether two knots are the same or different. The simplest knot is the unknot or trivial knot, which can be represented by a loop with no crossings. S1 r3 picture in the plane or slide diagram with crossing tame knots finite number of arcs only two strands at a crossing nice invariant property. Thurston showed that a knot complement will either be seifert bered.
When theres more than one loop, its called a link and the individual loops are called components of the link. Kauffman, is a professor in the department of mathematics, statistics, and computer science at the university of illinois at chicago. What is the importance of knot theory to category theory. This is why the reef knot should never be used as a bend to join two ropes. Free pdf download customise fifa gt edit teams gt change squads rosters gt download updates. Introduction to knot theory chris john february, 2016 supervised by dr. Reidemeister and alexander around about 1930 were able to make signi cant progress in knot theory. We present in this chapter chapter ii the history of ideas which lead up to the development of modern knot theory. Adams has also written a comic book about knot theory called why knot. A picture of a knot is called a knot diagram or knot projection. Table of contents for the handbook of knot theory william w. Knot theory consists of the study of equivalence classes of knots.
Chapter 1 a beginning for knot theory is very nice, it gives you a general flabour and taste of different elements that are used in knot theory like the definition of the redemeister moves, what is a link, the linking number, prime knots, pretzel knots, conway characterization of a knot through continued fractions etc. Thus a hyperbolic structure on a knot complement is a complete invariant of the knot. Radar target detection handbook of theory and practice pdf. We are more detailed when prexx century history is reported. The analysis of this earthquake curve can be considered as knot theory in the wide sense. Then, after defining an equivalence relation on all possible ways of factoring a knot, we will show that there is only one. One publication, journal of knot theory and its rami. Recent developments in knot theory make it an important chapter in physical mathematics. Thistlethwaite, editors 1 colin adams, hyperbolic knots 2 joan s. Two applications of elementary knot theory to lie algebras. Knot theory some formalism a knot is an injective map h. Although the subject matter of knot theory is familiar. Knot theory, 2004 a lot of material, but quite concise reidemeister. Seven of the eight 3dimensional geometries will not concern us here, but they jessica s.
It will show how knot theory originated from early attempts to apply the study of knots to other disciplines, and over time has emerged as a promising field of mathematical exploration in its own right. Using the notion of a seifert surface of a knot, we define a knots genus, an additive invariant which allows to prove the existence of prime knots. If we could completely understand hyperbolic structures on knot complements, we could completely classify hyperbolic knots. Tejas kalelkar 1 introduction knot theory is the study of mathematical knots. Peter guthrie tait was the first person to make charts describing mathematical knots in the 1860s related pages.
Classical knot theory the study of knots as objects in their own righthas taken great strides, documented throughout this handbook see the contributions by birman and brendle, hoste, kau. This exploration of combinatorics and knot theory is geared toward advanced undergraduates and graduate students. The reader is referred to 30,31,39,40,4445 46 4756,75,76,79,80 for more information about relationships of knot theory with statistical mechanics, hopf algebras and quantum groups. Pdf this paper is an introduction to virtual knot theory and an exposition. Knots, quandles, fundamental groups, handles, knot colorings, symmetry, 5 surfaces, klein bottle, projective plane 6 1. A crossing a knot is in the threedimensional space and we think that it is made of a very thin string. Although the subject matter of knot theory is familiar to everyone and its problems are easily stated, arising not only in many branches of mathematics but also in such diverse. A knot is called prime if it can not be represented as a connected sum of two knots such that both of these are knotted. Its very humorous but is a genuine introduction to the mathematics involved. Downloadan introduction to knot theory lickorish pdf. A brief history of knot theory erin col berg abstract. In the rst chapter, we will see how the research on this subject changed during the time. Handbook of theory and practice covers a set of graphical solutions to the detection.
Category theory revolutionized knot theory by emphasizing that we should take knots and links and break them into little pieces called tangles. Knot theory is a kind of geometry, and one whose appeal is very direct because the objects studied are perceivable and tangible in everyday physical space. Pdf knot theory from the perspective of field and string. It is a consequence of hakens work that the genus of a knot is algorithmically computable, because a minimal genus seifert surface can be arranged to be a fundamental normal surface. With more recent times we are more selective, stressing developments related to jones type invariants of links. In the past two decades, knot theory in general has seen much progress and many changes.
How ever, thurston and gabai found another method for determining the genus of knots, by using the theory of taut foliations 97, 21. Many products that you buy can be obtained using instruction manuals. Start moving along the knot, as we encounter the crossings we start numbering them, so every crossing has two numbers assigned to it once we have traced the entire knot. Pdf introduction to virtual knot theory researchgate. Then, after defining an equivalence relation on all possible ways of. A place where parts of the loop cross over is called a crossing. In this dissertation, we are going to give a brief introduction of knot theory, looking at di erent aspects.
Inspired by lord kelvins vortex theory of the atom 59, the scottish physicist p. By powerful communication of ideas i actually mean something precise. Peter guthrie tait was the first person to make charts describing mathematical knots in the 1860s other pages. Classicalknot theory studies the position of a circle knot or of several circles link in r3 or s3 r3 the fundamental problem of classical knot theory is the classi. Peter guthrie tait was the first person to make charts describing mathematical knots in the 1860s. A braid is an em b ed d in g of a co llect ion of stran d s th at h ave th eir en d s.
A survey of knot theory, 1990 a lot of material, but quite concise v. Category theory can be quite technical and dry, and difficult to understand if you dont have the motivating examples in mind. Of course, from here it turns out that just as always in topology, where we have an obstruction theory we also have a classi cation theory given by changing dimensions just a bit. Knotting occurs in dna and there is interaction between knot theorists and biologists. Untill the early 1980s it was an area in the backwaters of topology. Im not entirely sure that you want to read a category theory book. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. Thousands of refereed articles about knots have been published during just the past ten years. It is a meeting ground of such diverse branches of mathematics as group theory, matrix theory, number theory, algebraic geometry, and differential. This paper will provide a chronological description of the development of knot theory. Pdf knot theory from the perspective of field and string theory.
Each arrow has a domain and a codomain which are objects. Scottish physics and knot theorys odd origins introduction. For a mathematician, a knot is a closed loop in 3dimensional space. Read online now knot theory ebook pdf at our library.
Handbook of knot theory this page intentionally left blank handbook of knot theoryeditorswilliam menasco depart. It is a meeting ground of such diverse branches of mathematics as group theory, matrix theory, number theory, algebraic geometry, and differential geometry, to name some of the more. It describes knot coloring 2 by quandles, the fundamental group of a knot complement, and handledecompositions of 3 knot complements. A brief incursion into knot theory trinity university. This paper is a very brief introduction to knot theory. The jones polynomial of an alternating link 41 exercises 48 chapter 6. The knot group of a knot awith base point b2s3 ima is the fundamental group of the knot complement of a, with bas the base point. Knot theory was a respectable if not very dynamic branch of topology until the discovery of the jones polynomial 1984 and its connections with physics speci cally, quantum eld theory, via the work of witten. Tait set out to systematically enumerate knots based on their crossing number.
Peter scott has written an excellent introduction to the eight 3dimensional geometries scott, 1983. Knot theory is one of the most active research areas of mathematics today. Theoremfreedman,gompf,morrison,walker if some knot k satisfies sk. A beginning for knot theory 1 exercises chapter 2. Knot theory simple english wikipedia, the free encyclopedia. We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. Get knot theory pdf file for free from our online library pdf file. Hoste, the enumeration and classification of knots and links, handbook of. We can compose tangles by sticking them together vertically in the. Interplay with 4manifold theory conjecturesmooth poincare conjecture in 4 dimension m4 with. In the early 1990s, knot invariants which encompass the jones polynomial and its generalizations, called the finite type invariants, were discovered by vassiliev and goussarov. As michael comments, colin adams has a well regarded text called the knot book.
Jaap van oosten department of mathematics utrecht university the netherlands revised, july 2002. Table of contents for the handbook of knot theory pitzer college. Knots theory and linear algebra sam nelson in this lecture, we will see some connections between my area of research, a sub eld of lowdimensional topology known as knot theory, and linear algebra. Knot theory was a respectable if not very dynamic branch of topology until the discovery of the jones polynomial 1984 and its connections with physics speci. We are now ready to combine the notions of retraction and deformation. Radar target detection handbook of theory and practice pdf handbook of theory and practice.
Motivation for such a topological study of knots is meant to come from a curiosity to know how the ge ometry of threedimensional space can be explored by knotting phenomena using precise mathematics. Purcell, hyperbolic knot theory, c 2017 may 30, 2017 iii. The nokia that everyone knows and cares about creates smartphones, and that is being purchased by microsoft. In knot theory, the ends of the rope are attached so that there is no possible way for the knot to be untied. Knot theory is a branch of topology that concerns itself with abstract properties of mathematical knots the spatial arrangements that in principle could be assumed by a closed loop of string. An introduction to knot theory and the knot group 5 complement itself could be considered a knot invariant, albeit a very useless one on its own. K1 k2 balreira trinity university knot theory major. A brief history of category theory 9 this course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between di. Seifert surfaces and knot factorisation 15 exercises 21 chapter 3.
Well start with some preliminaries about knots, then see. Moreover, its a large subject and youll only need the tools related to the part of knot theory you want to study. Also see the book applications of knot theory, by buck and flapan. The basic question one asks in knot theory is, given two knots how to know if they are the same knot or not.