Parity and relative parity in knot theory arxiv vanity. 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. Pdf this paper is an introduction to virtual knot theory and an. Unsolved problems in virtual knot theory and combinatorial knot. An immediate invariant that comes to mind is the topological space s3 nk, the complement of knots.
Manturov, for attention to my mathematical work during all my life. 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. Knot theory is a kind of geometry, and one whose appeal is very direct hecause the objects studied areperceivable and tangible in everydayphysical space. Parity in knot theory vassily o manturov embedding of compacta, stable homotopy groups of spheres, and singularity theory p m akhmetev. In the mathematical area of knot theory, a reidemeister move is any of three local moves on a link diagram. Close to what well cover in the first half of the course. This has been known thattwo knots are ambient isotopic if and only if their complements are isotopic. We still do not know whether the free knot whose gauss diagram is a. Manturov and others published knot theory find, read and cite all the research you need on researchgate. Vassily olegovich manturov at bauman moscow state technical university.
An introduction to knot theory, by raymond lickorish. Download a survey of hyperbolic knot theory temple university book pdf free download link or read online here in pdf. Read online a survey of hyperbolic knot theory temple university book pdf free download link book now. Pdf unsolved problems in virtual knot theory and combinatorial. This principle comes from a very easy formula k ke, where k on the lhs is a virtual or free knot i. Parity theory discovered by the secondnamed author leads to a new perspective in virtual knot theory, the theory of cobordisms in two. A reason why virtual knots are important, and a relation between qft quantum field theory and virtual knots. 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. Here is a link to the cdbook by chmutov, duzhin, and mostovoy.
The main objective in knot theory is to invent more and more powerful invariants. Some suggestions for reading about knots and links comment. 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. A survey of knot theory, 1990 a lot of material, but quite concise v.
An elementary introduction to the mathematical theory of knots colin c. Knot theory is notable not only for its expert presentation of knot theorys state of the art but also for its accessibility. But we have the following correspondence between flat virtual knots and signed chord diagrams. 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. 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.
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 by vassily olegovich manturov overdrive. The paper discusses uses of parity pioneered by vassily manturov and. This leads to some heuristic ideas how to construct quandles and extract virtual link invariants. For a mathematician, a knot is a closed loop in 3dimensional space. Dqgfreruglvpriiuhhnqrwv vassily o manturov to cite. 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. 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.
Vassily manturov the mathematics genealogy project. 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. Knot theory now plays a large role in modern mathematics, and the most signi cant results in this theory have been obtained in the last two decades. Virtual knots have many exciting connections with other fields of knots theory. In this chapter, we briefly explain some elementary foundations of knot theory. Modern research methods in knot theory can be moreorless grouped into several categories. Mathematics genealogy project department of mathematics north dakota state university p.
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. In the present work, we construct an invariant of virtual knots valued in infinitedimensional lie algebras and establish some properties of it. I personally found the focus on invariants very useful. Join researchgate to discover and stay uptodate with the latest research from leading experts in knot theory and many other scientific topics. 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. The references below all have their own references, that will take you in many directions. Markov theorem for free links journal of knot theory.
Using the parity property arising from gauss diagrams we show that even a gross simpli. The present monograph is devoted to lowdimensional topology in the context of two thriving theories. Knot theory, second edition is notable not only for its expert presentation of knot theorys state of the art but also for its accessibility. Pdf this paper is a concise introduction to virtual knot theory, coupled with a list of. It suffices to mention the great progress in knot homology theory khovanov homology and.
If a free knot diagram is complicated enough then it realizes itself. 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. The latter means that it appears as a subdiagram in any diagram equivalent to it. It is valuable as a professional reference and will serve equally well as a text for a course on knot theory. Thus, dehns theorem reduces the trivial link recognition problem to the free. 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 ofthe more prominent ones. The notion of free link is a generalized notion of virtual link. Journal of knot theory and its ramifications vol 21, no.
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. Jones polynomials and classical conjectures in knot theory. Of all the methods, these are most directly connected to the topology of the knot. The theory of free knots has been pursued with much energy by manturov and his collaborators. It suffices to mention the great progress in knot homology theory khovanov homology and ozsvathszabo heegaardfloer homology, the apolynomial which give rise. Manturov, parity and cobordisms of free knots to appear in. Virtual knots and infinitedimensional lie algebras. 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. The present volume grew out of the heidelberg knot theory semester, organized by the editors in winter 200809 at heidelberg university. A family of polynomial invariants for flat virtual knots. A simple invariant is constructed which obstructs a free knot to be truncated. 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. All books are in clear copy here, and all files are secure so dont worry about it.
The most important example of a knot theory with parity is the theory of virtual knots. Journal of knot theory and its ramifications vol 29, no 02. One of the most important reason why we need virtual knots is as follows. In particular, this invariant provides an obstruction to the. 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. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary threemanifold and classical knot theory. A reidemeister move is an operation that can be performed on the diagram of a knot whithout altering the corresponding knot. Elementary constructions of homfly and kau man polynomials l. New to this edition is a discussion of heegaardfloer homology theory and apolynomial of classical links, as well as updates throughout the text. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. The conference program of the rcckt2017 will include. The mathematics genealogy project is in need of funds to help pay for student help and other associated costs. Knot theory now plays a large role in modern mathematics, and the most signifi cant results.
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