In part ii the technique of 6jsymbols is used to define state sum invariants of 3 manifolds. Quantum invariants of knots and3 manifolds clement maria the university of queensland june 2015. Yokota, y kashaevs conjecture and the chernsimons invariants of knots and links. Abstracts for vorlesung invariants of knots and 3 manifolds alternatively have a look at the program. 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. Allorder volume conjecture for closed 3manifolds from complex.
Enter your mobile number or email address below and well send you a link to download the free kindle app. 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. Quantum invariants of links and 3valent graphs in 3manifolds. Quandle cocycle invariants for knots, knotted surfaces and. 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. 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.
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. State sum invariants of 3 manifolds and quantum 6jsymbols v. Arial symbol blackadder itc projekt domyslny microsoft equation 3. Quantum supergroups and topological invariants of three manifolds r. This paper provides a criterion for periodicity of rational homology threespheres. Quantum invariants of knots and 3 manifolds 3rd edition by vladimir g. Witten e quantum field theory and the jones polynomial. Inventiones mathematicae volume 103 1991 issue article nutzungsbedingungen digizeitschriften e. Computation of zkm or z1m is generally a hard problem. Holomorphic disks and topological invariants for closed three. Abstracts for vorlesung invariants of knots and 3manifolds. 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.
Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. 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. Ribbon categories nicely fit the theory of knots and links in s3. Holomorphic disks and threemanifold invariants 1163 we conclude with two applications. Jan 01, 2003 we consider quantum invariants of 3 manifolds associated with arbitrary simple lie algebras. 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. Quantum invariants of 3manifolds and poincar\e conjecture. 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. Quantum invariants of knots and 3manifolds lixpolytechnique.
Or can there be manifolds not related to spacetime so can quantum on manifold make sense. Recent interest in torsions is due to their connections with the seibergwitten invariants of 4 manifolds and the floertype homology of 3 manifolds. 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. Quantum invariants of knots and 3manifolds pdf free download. Quantum invariants of 3manifolds, tqfts, and hopf monads. Their relation to the tqfts constructed in part i is established via the theory of shadows. Approximating turaevviro 3manifold invariants is universal. Thus approximating certain turaevviro invariants of manifolds presented by heegaard splittings is a universal problem for quantum computation.
On complexity and turaev viro invariants of 3manifolds. Vesnin im sb ras complexity and tvinvariants september 1, 2016 1 30. This is a list of open problems on invariants of knots and 3 manifolds with expositions of their history, background, significance, or importance. Invariants of knots and 3manifolds from quantum groupoids. Holomorphic disks and topological invariants for closed threemanifolds by peter ozsvath.
Quantum invariants of knots and 3manifolds mathematical. For k small, there are relations between zkm and classical topological invariants of m. We give a table of calculations for genus 2 handlebody knots up to 6. 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 gives a first existence theorem of nontrivial finite type invariants for knots in closed irreducible 3 manifolds. Ohtsuki abstract this is a list of open problems on invariants of knots and 3manifolds with expositions of their history, background, signi. 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. Pdf invariants of knots and 3manifolds from quantum. On abelian quantum invariants of links in 3manifolds.
In part ii the technique of 6jsymbols is used to define state sum invariants of 3manifolds. Springer nature is making coronavirus research free. This establishes a novel relation between the task of distinguishing nonhomeomorphic 3 manifolds and the power of a general quantum computer. Jo in terms of topological quantum field theory, reshetikhin and turaev rt. This thesis provides a partial answer to a question posed by greg kuperberg in qalg9712003 and again by justin roberts as problem 12.
Invariants of 3manifolds via link polynomials and quantum groups. Quantum hyperbolic invariants of 3manifolds with psl2. Quantum invariants of knots and 3manifolds 3rd edition. Knot theory available for download and read online in other formats. This book provides an extensive and selfcontained presentation of quantum and related invariants of knots and 3manifolds. The torsions were introduced by kurt reidemeister in 1935 to give a topological classification of lens spaces. The content of this paper was exposed at the minisemester in knot theory in war. Request pdf quantum invariants of 3manifolds and poincar\e conjecture a new quantum gauge model is proposed. Jan 30, 2016 does the words quantum manifolds make any sense. Quantum groups and 3manifold invariants topological field. Quantum invariants and finite group actions on three. These invariants are linear sums of yokotas invariants for colored spatial graphs, which is defined by using kauffman bracket. Two fundamental constructions of 3dimensional tqfts, which give rise in particular to scalar invariants of closed 3manifolds, are due to reshetikhinturaev. It turns out that in order to describe the spaces em we have to deal with the following problem.
Part iii provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3 space. Turaev invariants of 3manifolds, called also the quantum invariants, extend to. Jan 20, 2003 the aim of the lectures is to survey the theory of torsions of 3 dimensional manifolds. Perturbative invariants of 3manifolds with the rst betti. Let us, now, briefly describe the main ideas of the paper. From this quantum gauge model we derive a quantum invariant of 3manifolds. The construction is in the spirit of topological quantum eld theory and the invariant is calculated from a triangulation of the 3 manifold.
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