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Author Przytycki, Józef H.
Source CiteSeerX
Content type Text
File Format PDF
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Algebra Situs ♦ Ambient Isotopy Class ♦ Teacher Joan Birman ♦ Statistical Mechanic ♦ Polynomial Invariant ♦ Many Ingredient ♦ Lie Algebra ♦ Planar Algebra ♦ Main Object ♦ Q-homotopy Skein Module ♦ State Model ♦ Quantum Invariant ♦ Algebraic Topology ♦ Algebra Structure ♦ Skein Module ♦ Simple Example ♦ Free Module ♦ Th Birthday ♦ Quantum Group ♦ Non-commutative Geometry ♦ Algebraic Structure ♦ Algebraic Geometry ♦ Jones Construction
Description Dedicated to my teacher Joan Birman on her 70’th birthday. Algebra Situs is a branch of mathematics which has its roots in Jones ’ construction of his polynomial invariant of links and Drinfeld’s work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and statistical mechanics. Algebraic topology based on knots may be characterized as a study of properties of manifolds by considering links (submanifolds) in a manifold and their algebraic structure. The main objects of the discipline are skein modules, which are quotients of free modules over ambient isotopy classes of links in a manifold by properly chosen local (skein) relations. We concentrate, in this lecture, on one relatively simple example of a skein module of 3-manifolds – the q-homotopy skein module. This skein module already has many ingredients of the theory: algebra structure, associated Lie algebra, quantization, state models... 1
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article
Publisher Date 1998-01-01
Publisher Institution Proceedings of the Conference in Low-Dimensional Topology in Honor of Joan Birman’s 70th Birthday, Columbia University/Barnard