Perturbative invariants of 3manifolds with the rst betti number 1 tomotada ohtsuki abstract it is known that perturbative invariants of rational homology 3spheres can be formulated by using arithmetic perturbative expansion of quantum invariants of them. Problems on invariants of knots and 3manifolds preface arxiv. This establishes a novel relation between the task of distinguishing nonhomeomorphic 3 manifolds and the power of a general quantum computer. Problems on invariants of knots and 3manifolds edited by t. Perturbative invariants of 3manifolds with the rst betti. Ohtsuki preface the workshop and seminars on \invariants of knots and 3manifolds was held at research institute for mathematical sciences, kyoto university in september 2001.
M is countable and the range has the cardinality of the reals. Quantum invariants of 3manifolds, tqfts, and hopf monads. Part iii provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3 space. Or do you automatically equate manifolds to lorentzian manifold and it becomes a problem of quantum gravity. Arial symbol blackadder itc projekt domyslny microsoft equation 3. On abelian quantum invariants of links in 3manifolds. On complexity and turaev viro invariants of 3manifolds andrei vesnin sobolev institute of mathematics, novosibirsk, russia xix geometrical seminar zlatibor, serbia, august 28 september 4, 2016 a. Their relation to the tqfts constructed in part i is established via the theory of shadows. Part iii provides constructions of modular categories, based on.
For a general 3manifold it is hopeless to give nice formulae for wrtinvariants. Inventiones mathematicae volume 103 1991 issue article nutzungsbedingungen digizeitschriften e. Algebraic invariants of links and 3 manifolds trondheim, december 2015 masters thesis masters thesis trondheim, 2015 ntnu norwegian university of science and technology faculty of information technology, mathematics and electrical engineering department of mathematical sciences. In part ii the technique of 6jsymbols is used to define state sum invariants of 3 manifolds. Quantum supergroups and topological invariants of three manifolds r. Quasiquantum groups, knots, threemanifolds, and topological field. We show how to construct, starting from a quasihopf algebra, or quasiquantum group, invariants of knots and links. Meyer knot invariants and the thermodynamics of lattice gas automata free download as pdf file.
We use the categories of representations of finitedimensional quantum groupoids weak hopf algebras to construct ribbon and modular categories that give rise to invariants of knots and 3 manifolds. Invariants of 3manifolds via link polynomials and quantum. Msri workshop schedules quantum invariants of links and. Quantum invariants of knots and 3manifolds mathematical. Keywords invariant, knot, 3manifold, jones polynomial, vassiliev in variant, kontsevich. This is a list of open problems on invariants of knots and 3manifolds with expositions of their history, background, significance, or importance. Quantum supergroups and topological invariants of three. Quantum groups and 3manifold invariants topological. This algebraic framework consists of a tensor category with a condition on the duals which we. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. Jan 01, 2003 we consider quantum invariants of 3 manifolds associated with arbitrary simple lie algebras. Msri has been supported from its originsby the national science foundation,now joined by the national security agency,over 100 academic sponsor departments,by a range of private foundations, and by generous and farsighted individuals. Quantum hyperbolic invariants of 3manifolds with psl2.
We use the categories of representations of finitedimensional quantum groupoids weak hopf algebras to construct ribbon and modular categories. Quantum groups and 3manifold invariants topological field. Msri is a 501c 3 taxexempt organization and your donation is taxdeductible within the guidelines of u. The wittenreshetikhinturaev invariants of 3manifolds, called also the quantum invariants, extend to a topological quantum field theory tqft in dimension 3. Ribbon categories nicely fit the theory of knots and links in s3. Invariants of 3manifolds associated with quantum groups. In part ii the technique of 6jsymbols is used to define state sum invariants of 3manifolds.
Quantum invariants of links and 3valent graphs in 3manifolds. We give a table of calculations for genus 2 handlebody knots up to 6. Here we list related integrality results for quantum invariants for closed 3manifolds. Invariants of knots and 3manifolds from quantum groupoids. These invariants are linear sums of yokotas invariants for colored spatial graphs, which is defined by using kauffman bracket. Finally, we will describe the modern, highercategorical perspective on tqfts that includes not only invariants of 2 and 3manifolds but also algebraic data associated to. This establishes a novel relation between the task of distinguishing nonhomeomorphic 3manifolds and the power of a.
This paper presents an algebraic framework for constructing invariants of closed oriented 3manifolds by taking a state sum model on a triangulation. It turns out that in order to describe the spaces em we have to deal with the following problem. Invariants of 3 manifolds via link polynomials and quantum groups. In the mathematical theory of knots, the kontsevich invariant, also known as the kontsevich integral of an oriented framed link, is a universal vassiliev invariant in the sense that any coefficient of the kontsevich invariant is of a finite type, and conversely any finite type invariant can be presented as a linear combination of such coefficients. Quantum invariants of knots and 3 manifolds by ronreeves issuu. In particular, the compatibility con dition between multiplication and comultiplication of a biajgebra corresponds to a fiplitting move of templates along branch lines. Ams transactions of the american mathematical society. G invariants of 3manifolds via link polynomials and quantum groups. We prove that the quantum so3invariant of an arbitrary 3manifold m is always an algebraic integer if the order of the quantum parameter is coprime with the order of the torsion part of h1m,z. Quantum invariants of knots and 3 manifolds pdf bulletin new series of the. Part iii provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3space.
The lift of the colored jones polynomial is given by the witten reshetikhinturaev wrt invariant which associates with any closed oriented 3manifold, a semisimple lie algebra and a root of unity a complex number 28. Mar 30, 2017 finally, we will describe the modern, highercategorical perspective on tqfts that includes not only invariants of 2 and 3 manifolds but also algebraic data associated to manifolds of every. Zhang department of pure mathematics university of adelaide adelaide, sa 5001, australia february 1, 2008 the reshetikhin turaeve approach to topological invariants of three manifolds is generalized to quantum supergroups. Invariants of 3manifolds via link polynomials and quantum groups. We consider quantum invariants of 3manifolds associated with arbitrary simple lie algebras. Easily share your publications and get them in front of issuus. First, the manifold m can be obtained from s3 by integer surgery around a 2. Perturbative invariants of 3 manifolds with the rst betti number 1 tomotada ohtsuki abstract it is known that perturbative invariants of rational homology 3 spheres can be formulated by using arithmetic perturbative expansion of quantum invariants of them. State sum invariants of 3 manifolds and quantum 6jsymbols v. Preface a fundamental problem with quantum theories of gravity, as opposed to the other forces of nature, is that in ei. Quantum invariants of knots and 3manifolds vladimir g. We also discuss how to set the volume conjecture for the coloured jones invariants jnl of hyperbolic knots l in.
Ohtsuki preface the workshop and seminars on invariants of knots and 3manifolds was held at research institute for mathematical sciences, kyoto university in september 2001. In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored jones polynomial. Quantum invariants of knots and3 manifolds clement maria the university of queensland june 2015. Unified quantum invariants for integral homology spheres. Quantum invariants of knots and 3manifolds and millions of other books are available for amazon kindle. Holomorphic disks and topological invariants for closed.
Over 10 million scientific documents at your fingertips. We also discuss how to set the volume conjecture for the coloured jones invariants jnlof hyperbolic knots l in s3 in the framework of the general qhi theory. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. Holomorphic disks and topological invariants for closed three. Holomorphic disks and topological invariants for closed threemanifolds by peter ozsvath. Quantum invariants of knots and 3manifolds pdf free download. G invariants of 3 manifolds via link polynomials and quantum. Algebraic invariants of links and 3manifolds trondheim, december 2015 masters thesis masters thesis trondheim, 2015 ntnu norwegian university of science and technology faculty of information technology, mathematics and electrical engineering department of. Two fundamental constructions of 3dimensional tqfts, which give rise in particular to scalar invariants of closed 3manifolds, are due to reshetikhinturaev. Turaev some quite amazing results have appeared in the last two decades that connect two seemingly different fields of knowledge, namely topology and quantum field theory. This thesis provides a partial answer to a question posed by greg kuperberg in qalg9712003 and again by justin roberts as problem 12. In hindsight, the discovery in 1984 of the jones polynomial was the beginning of the new subject of quantum. Holomorphic disks and threemanifold invariants 1163 we conclude with two applications. There were 25 talks in the workshop in september 1721, and there were 27 talks in the.
For k small, there are relations between zkm and classical topological invariants of m. Quandle cocycle invariants for knots, knotted surfaces and. Quantum invariants of knots and 3manifolds 3rd edition by vladimir g. The aim of the present paper is to give an, as general as possible, method of constructing quantum invariants of 3manifolds starting from a ribbon category or a ribbon hopf algebra. In particular, the jones polynomial 12 and its generalizations 6, can be obtained that way. Invariants of 3manifolds and projective representations of mapping. This gives a vast class of knot invariants and 3manifold invariants as well as a class of linear. Quantum invariants of knots and 3 manifolds 3rd edition by vladimir g. Computation of zkm or z1m is generally a hard problem. Quantum invariants of knots and 3manifolds 3rd edition. The aim of the lectures is to survey the theory of torsions of 3dimensional manifolds.
In some cases, these invariants give rise to invariants of the threemanifolds. A study of knots, 3manifolds, and their sets series on knots and everything. The torsions were introduced by kurt reidemeister in 1935 to give a topological classification of lens spaces. Such relations between diagrams and algebras are chfllaeteristic in qlhultum inwlliants of knots and 3manifolds. An invariant of closed oriented 3manifolds is constructed. Let us, now, briefly describe the main ideas of the paper. On complexity and turaev viro invariants of 3manifolds. Approximating turaevviro 3manifold invariants is universal. Pdf invariants of knots and 3manifolds from quantum. In the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored jones polynomial of surgery presentations of the knot complement. This is a list of open problems on invariants of knots and 3 manifolds with expositions of their history, background, significance, or importance. Thus approximating certain turaevviro invariants of manifolds presented by heegaard splittings is a universal problem for quantum computation.
This paper presents an algebraic framework for constructing invariants of closed oriented 3 manifolds by taking a state sum model on a triangulation. Recent interest in torsions is due to their connections with the seibergwitten invariants of 4manifolds and the floertype homology of 3manifolds. This gives a first existence theorem of nontrivial finite type invariants for knots in closed irreducible 3 manifolds. Save up to 80% by choosing the etextbook option for isbn. Quantum invariants of knots and 3 manifolds by ronreeves. Vesnin im sb ras complexity and tvinvariants september 1, 2016 1 30. Jan 30, 2016 does the words quantum manifolds make any sense.
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