Knot theory week 1 ucsb 2015 1 introduction outside of mathematics, knots are ways to loop a single piece of string around itself. 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. Dover modern math originals rediscovering northwest denver. The basic question one asks in knot theory is, given two knots how to know if they are the same knot or not.
The bridge number bl of a link lis the smallest integer nsuch that lmay be expressed as the tangle sum of two trivial nstring tangles. Read an interactive introduction to knot theory by inga johnson,allison k. For example, consider the following knot projections. An interactive introduction to knot theory by inga johnson and allison k henrich topics. Read an interactive introduction to knot theory by inga johnson available from rakuten kobo. Recall that a graph consists of a set of vertices and a set of edges that connect them. If both ends of the line will be loaded, then the clove hitch will. This wellwritten and engaging volume, intended for. The focus of this article is this elementary aspect to knot theory. We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. This book was written as an introductory text for a onesemester course and, as such, it is far from a comprehensive reference work. This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial.
Sections 6 and 7 provide an introduction to vassiliev invariants and the remarkable relationship between lie algebras and knot theory. Mathematics in life including gambling and games is described and. Introduction this survey considers three current areas of study that combine the elds of graph theory and knot theory. Knots can be studied at many levels and from many points of view. Classification and detection of knots concepts from the mathematical field of knot theory have. 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.
Spectacular graphics make this online game instructional and fun. The fishermans knot is similar but easier to tie with cold, wet hands. I describe the homologies of covering spaces of a knot, the alexander polynomial, the homfly polynomial, the colored jones polynomial, and the volume conjecture. Ropelength and tight knots jason cantarella university of georgia ictp knot theory summer school, trieste, 2009. Knot theory this chapter looks at some of the fundamental building blocks of knot theory. An interactive introduction to knot theory inga johnson, allison k. A polygonal knot is a knot whose image in r 3 is the union of a finite set of line segments. 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. Interactive introduction to knot theory by allison k. Lecture notes class projects course information this is an introductory course in knot theory. Seifert surfaces and knot factorisation 15 exercises 21 chapter 3. We can visualize a mathematical knot as a knot that we would encounter in our everyday experience, but with the loose ends glued together.
A beginning for knot theory 1 exercises chapter 2. An introduction to the theory of knots computer graphics. Its very humorous but is a genuine introduction to the mathematics involved. Dover modern math originals paperback january 18, 2017 by inga johnson author, allison k.
Introduction knot theory as we know it first gained prominence as a. An elementary introduction to the mathematical theory of knots. An interactive introduction to knot theory by inga johnson. You can hand in your solutions in the letter box in hg f 28. Although beyond the scope of this leisurely introduction. Gives good explanations of many of the relevant concepts, but does not cover all the material of this course. An introduction to knot theory matt skerritt c9903032 june 27, 2003 1 introduction a knot, mathematically speaking, is a closed curve sitting in three dimensional space that does not intersect itself. An introduction to knot theory university of newcastle.
Jump right into the action on the interactive search for a master of knot disguise. This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in threedimensional space. An interactive introduction to knot theory mathematical. Introduction in the past 50 years, knot theory has become an extremely welldeveloped subject. Its history, its people, its landmarks hoods texas brigade, its marches, its. Dover modern math originals kindle edition by johnson, inga, henrich, allison k download it once and read it on your kindle device, pc, phones or tablets. Trefoil knot fi g ure 4 c lo sing b ra ids to fo rm k no ts a nd link s. Feb 24, 2017 interactive introduction to knot theory by allison k. Theory and its applications knot theory mathematical association of america textbooks an interactive introduction to knot theory aurora. In mathematics, we mean something slightly di erent by a knot. Our exploration of the history of knot theory will conclude with a discussion of current conjecture regarding possible applications and the direction of the future of knot theory. Tejas kalelkar 1 introduction knot theory is the study of mathematical knots. A special case of this problem is one of the fundamental questions of knot theory. Knots which are not tame are called wild, and can have pathological behavior.
See all 2 formats and editions hide other formats and editions. An interactive introduction to knot theory an interactive introduction to knot theory inga johnson and allison k. Jan 04, 2011 this paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. In my opinion it works well after calculus, linear algebra, andor differential equations, but technically does not require any knowledge on any of those subject matters. The handson approach features many exercises to be completed by readers. This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Intuitively if we were to take a piece of string, cord, or the like, tie a knot in it and then glue the loose ends together, we would have a knot. The elementary ideals 72 72 78 86 87 90 94 94 96 100 101 chapter viii the knot polynomials introduction.
Read an interactive introduction to knot theory online by inga. Geometry of alternating links 32 exercises 40 chapters. This is an introduction to knot theory for nonspecialists, focused on quantum invariants. A quantum introduction to knot theory hitoshi murakami abstract. We are the home of such worldclass theorists as paul j. An introduction to knot theory these notes were written. The central problem of knot theory is determining whether two knots can be rearranged without cutting to be exactly alike. Buy ebook an interactive introduction to knot theory by inga johnson, allison k. Chapter vii the free calculus and the elementary ideals introduction. An interactive introduction to knot theory an interactive introduction to knot theory inga johnson and allison henrich. For flat material such as seatbelts, the water knot is best.
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. Of all the technical areas in which we publish, dover is most recognized for our magnificent mathematics list. Polyhedral decomposition of the figure8 knot complement 2. Knot theory consists of the study of equivalence classes of knots. This wellwritten and engaging volume, intended for undergraduates, introduces knot theory, an area of growing interest in contemporary mathematics. Although the subject matter of knot theory is familiar. This account is an introduction to mathematical knot theory, the theory of knots. Historically, mathematicians have studied various graph. A spatial embedding of a graph is, informally, a way to place the graph in space. A framed knot is the extension of a tame knot to an embedding of the solid torus d 2.
A tame knot is any knot equivalent to a polygonal knot. Cohen set theory and the continuum hypothesis, alfred tarski undecidable theories, gary chartrand introductory graph theory, hermann weyl the concept of a riemann surface, shlomo sternberg dynamical systems. In knot theory and 3manifold theory, often the adjective tame is omitted. Novel aspects of knotplot include interactive construction and manipulation of a large variety of knots and their embeddings into threespace, automated and semiautomated knot relaxation and simplification algorithms, and techniques for calculating and converting among a number of mathematical.
A knot is any closed nonselfintersecting loop embedded in three dimensions. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916. Interactive knot theory with knotplot springerlink. Henrich, 9780486804637, available at book depository with free delivery worldwide. Henrich is the coauthor of a book on knot theory, an interactive introduction to knot theory with inga johnson, dover publications, 2017. Introduction knot theory is the study of the various properties and behaviors of mathematical knots.
The jones polynomial of an alternating link 41 exercises 48 chapter 6. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations. An introduction to knot theory with 114 illustrations springer. We describe the computer program knotplot, a topological drawing tool for knots and links.
Inga johnsons wonderful an interactive introduction to knot theory johnson, 2017. An interactive introduction to knot theory ebook by inga johnson. The majority of the pictures were drawn using inkscape or tikz, with a few exceptions taken from adams, 2004. An interactive introduction to knot theory inga johnson. She also coauthored the book a mathematicians practical guide to mentoring undergraduate research with michael dorff and lara pudwell, mathematical association of america, american mathematical society. Knotsintroduction wikibooks, open books for an open world. An interactive introduction to knot theory dover publications. Buy an interactive introduction to knot theory aurora. The opening chapter offers activities that explore the world of knots and linksincluding games with knots and invites the reader to generate their own questions in knot theory.
Polyhedral decomposition of the figure8 knot complement 9. This comic book comes with a plastic rope that can be knotted, unknotted, and twisted into different shapes. This decomposition appears in thurstons notes 19, and with a. Adams has also written a comic book about knot theory called why knot.
Henrich, ebook format, from the dymocks online bookstore. The merit of knot theory for the study of dimensional manifolds consists, first of all, in that every closed oriented dimensional manifold can be represented as a covering of the sphere, ramified over a certain link alexanders theorem. Henrich this wellwritten and engaging volume, intended for undergraduates, introduces knot theory, an area of growing interest in contemporary mathematics. The paper is relatively selfcontained and it describes virtual knot theory both combinatorially and in terms of the knot theory in thickened surfaces. Some of the material will be used later in this report, while some is included simply to give an idea of di erent techniques used in knot theory. Use features like bookmarks, note taking and highlighting while reading an interactive introduction to knot theory aurora.
The framing of the knot is the linking number of the image of the ribbon i. The ideas used in this chapter can be found in most introductory books or courses on. Introduction to knot theory chris john february, 2016 supervised by dr. Ropelength minimizers called tight knots exist in each knot and link type and are c1,1. Cornell robotic construction laboratorys log knot 1. I took intro to knot theory as a 300 level course junior level. But there remain several notoriously intractable problems about knots and links, many of which are surprisingly easy to state. 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. The knot book freeman and company, 2001 an easy to read elementary introduction to knot theory. I describe the homologies of covering spaces of a knot, the alexander polynomial, the homfly polynomial. Jan 04, 2011 this paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial. Prerequisites are only a basic familiarity with linear algebra and a willingness to explore the subject in a handson manner. Prerequisites are only a basic familiarity with linear algebra and a.1533 792 267 945 1259 1430 1236 1552 261 1472 419 1374 1552 757 860 1054 771 1366 117 1066 1300 1377 1209 533 1052 1383 614 929 360 478 253 1475 1216 1051 795 1270 115 1260 1256 412 65 1341 1168 282 469 1473