Jackson, iain moffatt this book provides an accessible introduction to knot theory, focussing on vassiliev invariants, quantum knot invariants constructed via representations of quantum groups, and how these two apparently distinct theories come together through the kontsevich. Quantum invariants of 3manifolds, tqfts, and hopf monads. Perturbative invariants of 3manifolds with the rst betti. We consider quantum invariants of 3manifolds associated with arbitrary simple lie algebras. The aim of this meeting is to introduce the theory of quantum groups and their representations, and to investigate associated 3dimensional topological quantum field theories tqfts. In this section, we recall some known definitions and facts that will be used in the rest of the paper. Approximating turaevviro 3manifold invariants is universal.
A useful tool here would be a measure ofcomplexityof a 3manifold. This establishes a novel relation between the task of distinguishing nonhomeomorphic 3manifolds and the power of a general quantum computer. Possible universal quantum algorithms for generalized. Amenable discrete quantum groups tomatsu, reiji, journal of the mathematical society of japan, 2006. In part ii the technique of 6jsymbols is used to define state sum invariants of 3 manifolds. Turaev, invariants of 3manifolds via link poly nomials and quantum graups, invent. Computing turaevviro invariants for 3manifolds springerlink. Mar 23, 2009 2 links, ospines, and turaevviro invariants of 3manifolds. Part iii provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3 space. The main part of the program of the masterclass consisted of the following lecture series by chris douglas and andre henriques title. Jo in terms of topological quantum field theory, reshetikhin and turaev rt.
Theorem reshetikhin,turaev aribbon category associates to everyvcoloured ribbon diagram a. Pdf invariants of knots and 3manifolds from quantum. Save up to 80% by choosing the etextbook option for isbn. We give a simple surface interpretation for each summand in the evaluation of turaevviro invariants, for the case of small up to eighth roots of unity.
The rtconstruction is widely viewed as a mathematical realization of wittens chernsimons tqft, see wi. Turaev invariants of 3manifolds, called also the quantum invariants, extend to. Polynomial invariants of knots, such as the jones and alexander polynomials, are constructed as quantum invariants, i. The monograph gives a systematic treatment of 3dimensional topological quantum field theories tqfts based on the work of the author with n. Introduction as an application we obtain the 0 1997 elsevier science ltd. Turaev, quantum invariants of knots and 3manifolds, w. We calculate the rtinvariants of all oriented seifert manifolds directly from. On complexity and turaev viro invariants of 3manifolds.
Part iii provides constructions of modular categories, based on quantum groups. Their relation to the tqfts constructed in part i is established via the theory of shadows. Z,m, l of embeddings of links l in 3 manifolds m, known as the witten reshetikhin turaev invariant. The data for the construction of the invariant is a tensor category with a condition on the duals, which we have called a spherical category. Extensive tables relative to a rich variety of 3manifolds are explicitly presented.
We consider certain invariants of links in 3manifolds, obtained by a specialization of the turaevviro invariants of 3manifolds, that we call colored turaevviro invariants. Pdf invariants of knots and 3manifolds from quantum groupoids. We calculate the rtinvariants of all oriented seifert manifolds directly from surgery presentations. Quantum invariants of knots and 3manifolds avaxhome. Vladimir georgievich turaev, born in 1954 is a russian mathematician, specializing in topology turaev received in 1979 from the steklov institute of mathematics his candidate of sciences degree phd under oleg viro. Turaev was a professor at the university of strasbourg and then became a professor at indiana university. This book provides an extensive and selfcontained presentation of quantum and related invariants of knots and 3manifolds. We work in the general framework of an arbitrary modular category as in v. Temperleylieb recoupling theory and invariants of 3.
We construct quantum hyperbolic invariants qhi for triples w. January 28, 2020 61 views 4 months ago unified wittenreshetikhinturaev invariants of rational homology 3spheres date. This gives a vast class of knot invariants and 3 manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. Ribbon categories nicely fit the theory of knots and links in s3.
The construction is in the spirit of topological quantum eld theory and the invariant is calculated from a triangulation of the 3manifold. The content of this paper was exposed at the minisemester in knot theory in war. Turaev, invariants of 3 manifolds via link poly nomials and quantum graups, invent. Once one has constructed a lot of 3manifold invariants, the question is to. Such invariants of framed links also give rise to invariants of 3manifolds via the dehn surgery construction. This gives a vast class of knot invariants and 3manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. Quantum groups and knot invariants, smf panoramas et syntheses 1997. Msri workshop schedules quantum invariants of links and. Quantum invariants of knots and 3 manifolds 3rd edition by vladimir g.
Wittenreshetikhin turaev invariants of the double of m. These are the wittenreshetikhin turaev and the turaev viro invariants. This book offers a selfcontained account of the 3 manifold invariants arising from the original jones polynomial. 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. Those seeking only some exposure to quantum topology, would do well to focus on the introduction and chapters ii. These are the wittenreshetikhinturaev and the turaevviro invariants. Quantum invariants of knots and 3manifolds pdf free download. From this interpretation follows an efficient scheme to compute these invariants. Starting from the kauffman bracket model for the jones polynomial and the diagrammatic temperleylieb algebra, higherorder polynomial invariants of links are. All the theory of quantum groups gives a systematic way of producing families of polynomial.
Volume 127, issues 12, 1 january 2003, pages 91123. Computation of zkm or z1m is generally a hard problem. Abstract this is a list of open problems on invariants of knots and 3manifolds with expositions of their history, background, signi. Author links open overlay panel dmitri nikshych a vladimir turaev b. An introduction to quantum and vassiliev knot invariants. Invariants of knots and 3manifolds from quantum groupoids. Reshetikhinturaev invariants of framed links and 3manifolds eugene rabinovich today i will tell as much as i can about the invariants reshetikhin and turaev constructed based on ed wittens seminal physics explorations with quantum chernsimons theory. Thus approximating certain turaev viro invariants of manifolds presented by heegaard splittings is a universal problem for quantum computation. Reshetikhinturaev invariants of seifert 3manifolds and. This list was made by editing open problems given in problem sessions in the workshop and seminars on invariants of knots and 3. This list was made by editing open problems given in problem sessions in the workshop and seminars on invariants of knots and 3manifolds held at kyoto in 2001. Invariants of 3 manifolds via link polynomials and quantum groups. Quantum invariants of knots and 3manifolds by vladimir g.
The monograph gives a systematic treatment of 3 dimensional topological quantum field theories tqfts based on the work of the author with n. We study the unified wittenreshetikhinturaev invariant for the brieskorn homology sphere. On colored turaevviro invariants for links in arbitrary 3. The tvconstruction is closely related to the ponzanoregge state sum model for 3dimensional quantum gravity, see ca. Computations of turaevviroocneanu invariants of 3manifolds. These invariants are based on the faddeevkashaevs quantum dilogarithms, interpreted as. An emergent trend in quantum computation is the topological quantum computation tqc. Z,m, l of embeddings of links l in 3manifolds m, known as the witten reshetikhinturaev invariant. Reshetikhinturaev invariants of seifert 3manifolds and a. Viro the ideas outlined above lead not only to numerical invariants of 3manifolds but rather to a 3dimensional nonoriented topological quantum. Mathematical society of japan memoirs project euclid.
Quantum invariants of knots and 3manifolds vitalsource. Reshetikhinturaev invariants of seifert 3manifolds. Yokota, on the volume conjecture for hyperbolic knots, preprint, arxiv. In part ii the technique of 6jsymbols is used to define state sum invariants of 3manifolds. Rozansky, wittens invariants of rational homology spheres at prime values of k and trivial connection contribution, commun. This book offers a selfcontained account of the 3manifold invariants arising from the original jones polynomial. An introduction to quantum groups lectures at ncgoa07 christian kassel summary. Invariants of 3manifolds via link polynomials and quantum. For this we adapt the argument of 2 to the case that ris odd and ais a primitive 2rth root of unity. These will be an isotopy invariant of framed links.
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. Via this correspondence and the dehn surgery formula, we compute turaev viroocneanu invariants from several subfactors for basic 3 manifolds including lens spaces and brieskorn 3 manifolds by using izumis data written in terms of sectors. Thus approximating certain turaevviro invariants of manifolds presented by heegaard splittings is a universal problem for quantum computation. Its treatment of modular categories, of modular functors, and of tqft has stood the test of. Such invariants of 3manifolds are known as wittinreshetikhinturaev invariants wrtinvariants. Examples edit let a \displaystyle a be a ribbon hopf algebra over a field k \displaystyle \bbbk one can take, for example, any quantum group over c \displaystyle \mathbb c.
Turaev, invariants of 3manifolds via link polynomials and quantum groups. Quantum invariants of knots and 3manifolds mathematical. For a general 3manifold it is hopeless to give nice formulae for wrtinvariants. Turaevinvariants of 3manifolds via link polynomials and quantum groups. In this paper, we consider pairs m, l, where m is an oriented closed 3manifold and l is an oriented link in m. Quantum hyperbolic invariants of 3manifolds with psl2. Their construction is based on a presentation of a pair m, l, where m is a closed oriented 3manifold, and is an oriented link, by a triangulation of m such that each component of l is an edge. Quantum invariants of knots and 3manifolds 3rd edition by vladimir g. An introduction to quantum and vassiliev knot invariants david m. In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the turaevviroocneanu tqft introduced by ocneanu and another comes from the sector theory introduced by izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjugate linear isomorphism preserving the products of the two algebras and. These invariants were discovered by nicolai reshetikhin and vladimir turaev in 1991, and were meant to be a mathematical realization of.
Quantum invariants of seifert 3manifolds and their. Quantum loewner evolution miller, jason and sheffield, scott, duke mathematical journal, 2016. Due to the strong appeal and wide use of this monograph, it is now available in its second revised edition. This subject was inspired by the discovery of the jones polynomial of knots and the wittenchernsimons field theory. Pushing a bit more the construction, we get the turaevviro invariant. Turaevviro invariants, colored jones polynomials and volume.
We consider quantum invariants of 3 manifolds associated with arbitrary simple lie algebras. 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 establishes a novel relation between the task of distinguishing nonhomeomorphic 3 manifolds and the power of a general quantum computer. 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. For k small, there are relations between zkm and classical topological invariants of m. Quantum invariants of knots and 3manifolds vladimir g. Quantum invariants of links and 3valent graphs in 3manifolds. A clear exposition of a less general form of the material in chapter xii can also be found in a number of papers of raymond lickorish. Part iii provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3space. Invariants of 3manifolds via link polynomials and quantum groups.
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