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Knot theory

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Knot theory

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The aim of this book is to present recent results in both theoretical and applied knot theory&#8212which are at the same time stimulating for leading researchers in the &#64257eld as well as accessible to non-experts. The book comprises recent research results while covering a wide range of di&#64256erent sub-disciplines, such as the young &#64257eld of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics.

Knot theory.

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These notes arise from lectures presented in Florence under the auspices of the Accadamia dei Lincee and deal with an area that lies at the crossroads of mathematics and physics. The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions. Professor Atiyah here presents an introduction to Witten's ideas from the mathematical point of view. The book will be essential reading for all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area.

Knot theory.

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The author constructs knot invariants categorifying the quantum knot variants for all representations of quantum groups. He shows that these invariants coincide with previous invariants defined by Khovanov for mathfrak{sl}_2 and mathfrak{sl}_3 and by Mazorchuk-Stroppel and Sussan for mathfrak{sl}_n. The author's technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is mathfrak{sl}_n, the author shows that these categories agree with certain subcategories of parabolic category mathcal{O} for mathfrak{gl}_k.

Knot theory.