Parity theory discovered by the secondnamed author leads to a new perspective in virtual knot theory, the theory of cobordisms in two. In particular, this invariant provides an obstruction to the. An elementary introduction to the mathematical theory of knots colin c. Knot theory now plays a large role in modern mathematics, and the most signifi cant results. The mathematics genealogy project is in need of funds to help pay for student help and other associated costs. This principle comes from a very easy formula k ke, where k on the lhs is a virtual or free knot i. Manturov, for attention to my mathematical work during all my life. Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and algebra. Of all the methods, these are most directly connected to the topology of the knot. Knot theory by vassily olegovich manturov overdrive. It is valuable as a professional reference and will serve equally well as a text for a course on knot theory. Join researchgate to discover and stay uptodate with the latest research from leading experts in knot theory and many other scientific topics.
The theory of free knots has been pursued with much energy by manturov and his collaborators. Dqgfreruglvpriiuhhnqrwv vassily o manturov to cite. The main objective in knot theory is to invent more and more powerful invariants. The most important example of a knot theory with parity is the theory of virtual knots. 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 the mathematical area of knot theory, a reidemeister move is any of three local moves on a link diagram. An introduction to knot theory, by raymond lickorish. In the present work, we construct an invariant of virtual knots valued in infinitedimensional lie algebras and establish some properties of it. Pdf introduction to virtual knot theory researchgate. This has been known thattwo knots are ambient isotopic if and only if their complements are isotopic. The contributed papers bring the reader up to date on the currently most actively pursued areas of mathematical knot theory and its applications in mathematical.
Close to what well cover in the first half of the course. Modern research methods in knot theory can be moreorless grouped into several categories. Parity and relative parity in knot theory arxiv vanity. If a free knot diagram is complicated enough then it realizes itself. The references below all have their own references, that will take you in many directions. In this chapter, we briefly explain some elementary foundations of knot theory. In, manturov showed that each flat virtual knot cannot be represented by a chord diagram, and introduced free knots which are a 11 correspondence with chord diagrams.
Readings for the nonspecialist a hoste, thistlethwaite and weeks, the first 1,701,936 knots, scienti. Vassily olegovich manturov at bauman moscow state technical university. In a unique presentation with contents not found in any other monograph, knot theory describes, with full proofs, the main concepts and the latest investigations in the field. Using the parity property arising from gauss diagrams we show that even a gross simpli. Knot theory, second edition is notable not only for its expert presentation of knot theory s state of the art but also for its accessibility. It suffices to mention the great progress in knot homology theory khovanov homology and ozsvathszabo heegaardfloer homology, the apolynomial which give rise. All books are in clear copy here, and all files are secure so dont worry about it.
If you would like to contribute, please donate online using credit card or bank transfer or mail your taxdeductible contribution to. The latter means that it appears as a subdiagram in any diagram equivalent to it. It suffices to mention the great progress in knot homology theory khovanov homology and. Vassily manturov the mathematics genealogy project. Thus, dehns theorem reduces the trivial link recognition problem to the free. Kurt reidemeister and, independently, james waddell alexander and garland baird briggs, demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three reidemeister moves. In this paper we define the group of free braids, prove the alexander theorem, that all free links can be obtained as closures of fr. Pdf this paper is an introduction to virtual knot theory and an. Knot theory, second edition is notable not only for its expert presentation of knot theorys state of the art but also for its accessibility. Mathematics genealogy project department of mathematics north dakota state university p.
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. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary threemanifold and classical knot theory. An immediate invariant that comes to mind is the topological space s3 nk, the complement of knots. 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. Manturov, parity and cobordisms of free knots to appear in. V o manturov over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and algebra. Since discovery of the jones polynomial, knot theory has enjoyed a virtual explosion of important results and now plays a significant role in modern mathematics. Read online a survey of hyperbolic knot theory temple university book pdf free download link book now. This leads to some heuristic ideas how to construct quandles and extract virtual link invariants. Jones polynomials and classical conjectures in knot theory. The notion of free link is a generalized notion of virtual link.
Pdf this paper is a concise introduction to virtual knot theory, coupled with a list of. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. For a mathematician, a knot is a closed loop in 3dimensional space. We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. Manturov and others published knot theory find, read and cite all the research you need on researchgate. Some suggestions for reading about knots and links comment. Elias gedney patron, east kingdom guild of st erasmus knots vocabulary where bend is a noun and middle is a verb bend a knot that is used to join two lines together bight the turn part of a loop fuse to partially melt the. Knot theory, 2004 a lot of material, but quite concise reidemeister.
One of the most important reason why we need virtual knots is as follows. Here is a link to the cdbook by chmutov, duzhin, and mostovoy. In this book we present the latest achievements in virtual knot theory including khovanov homology theory and parity theory due to v o manturov and graphlink theory. Get your kindle here, or download a free kindle reading app. A family of polynomial invariants for flat virtual knots. The present monograph is devoted to lowdimensional topology in the context of two thriving theories. Download a survey of hyperbolic knot theory temple university book pdf free download link or read online here in pdf. The paper discusses uses of parity pioneered by vassily manturov and. A reidemeister move is an operation that can be performed on the diagram of a knot whithout altering the corresponding knot. But we have the following correspondence between flat virtual knots and signed chord diagrams. Dear colleague, from monday july 3 to friday, july 7, 2017, at the bauman moscow state technical university is the 4th russianchinese conference on knot theory and related topics. Knot arithmetics3 torus knots4 fundamental group5 quandle and conways algebra6 kauffmans approach to jones polynomial7 jones polynomial. A reason why virtual knots are important, and a relation between qft quantum field theory and virtual knots. I personally found the focus on invariants very useful.
The mathematics of knots theory and application markus. A simple invariant is constructed which obstructs a free knot to be truncated. Elementary constructions of homfly and kau man polynomials l. We still do not know whether the free knot whose gauss diagram is a. Journal of knot theory and its ramifications vol 21, no. Knot theory is a kind of geometry, and one whose appeal is very direct hecause the objects studied areperceivable and tangible in everydayphysical space. 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. Journal of knot theory and its ramifications vol 29, no 02. This paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe. Unsolved problems in virtual knot theory and combinatorial knot.
Virtual knots have many exciting connections with other fields of knots theory. The present volume grew out of the heidelberg knot theory semester, organized by the editors in winter 200809 at heidelberg university. Knot theory is notable not only for its expert presentation of knot theorys state of the art but also for its accessibility. Khovanovs complex8 leerasmussen invariant, slice knots, and the genus conjecture ii theory of braids9 braids, links and representations10 braids and links11 algorithms of braid recognition12 markovs theorem. New to this edition is a discussion of heegaardfloer homology theory and apolynomial of classical links, as well as updates throughout the text. Using the parity property arising from gauss diagrams we show that even a gross simplification of the theory of virtual knots, namely, the theory of free knots, admits simple and highly nontrivial invariants. Markov theorem for free links journal of knot theory. Parity in knot theory vassily o manturov embedding of compacta, stable homotopy groups of spheres, and singularity theory p m akhmetev. A survey of knot theory, 1990 a lot of material, but quite concise v.
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