The Geometry and Physics of Knots
Abstract
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.
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The correct general axiomatic scheme for a physical quantum theory of gravity is Atiyah's set of axioms up to finite dimensionality of the Hilbert state space.
“These considerations leads us to propose that the correct general axiomatic scheme for a physical quantum theory of gravity is simply Atiyah's set of axioms [3] up to finite dimensionality of the Hilbert state space.”
From Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum-gravity model is the loop representation basis (1993) · cites this paper as [3]
The quantum states of the Ponzano-Regge theory have to be taken as quantum combinations Φ∆(c) of the colored triangulations induced on the 2-d boundary ∂M of the 3-d manifold M.
“The quantum states of the Ponzano-Regge theory have to be taken, following Atiyah's general formulations of topological quantum field theories, as quantum combinations Φ ∆ ( c ) of the colored triangulations (∆ , c ) induced on the 2-d boundary ∂M of the 3-d manifold M .”
From Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum-gravity model is the loop representation basis (1993) · cites this paper as [3]
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- Authors
- Atiyah, M.
- Affiliations
- University of Cambridge (GB)
- Corresponding author
- Atiyah, Michael
- Venue
- Cambridge University Press eBooks
- Publisher
- Cambridge University Press
- Published
- 1990-08-23
- Pages
- 88
- OpenAlex
- W1616784375
- Cited by
- 597
- References
- 0
- Citations by year
- 2026: 11 · 2025: 7 · 2024: 16 · 2023: 15 · 2022: 12 · 2021: 11 · 2020: 14 · 2019: 8 · 2018: 14 · 2017: 17 · 2016: 21 · 2015: 18 · 2014: 13 · 2013: 15 · 2012: 20
- FWCI
- 3.78
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Cambridge University Press eBooks (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- International Linear Collider (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic structures and combinatorial models
- Keywords
- knot theory, knot polynomials, topological field theory, gauge theory, mathematical physics
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