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- W73816557 abstract "In this paper we report three main results: (a) An algebraic-geometric construction of universal link invariants in quantum groups based on Reidemeister’s theorem; (b) A similar construction of universal tangle invariants -- diagrammatically a tangle is a link diagram with one external edge cut -- and a proof showing that, with a certain restriction on the quantum group, the set of all tangle invariants forms a subset of the centre of the quantum group; (c) A demonstration that a classical simple Lie (and Kac-Moody) algebra can be at least twice deformed to give a “twisted” quantum group with an associated twisted Hopf structure -- the classical Alexander-Conway link Polynomial is the simplest link invariant constructed from the twisted quantum group of (2). These results are used to derive a number of theorems on the quantum group invariants of tangles, knots and links and their representations. They are also used to give insight to our understanding of the relation between quantum groups and conformal and topological field theories. It appears that, effectively, only the maximum Abelian subalgebra of the Hopf structure of the quantum group acts in these theories. It is shown that a tangle to a link in knot theory is what a “Wilson tangle” is to a Wilson line in the Chern-Simons-Witten topological field theory. The tangle theorem derived in (b) asserts that the set of all Wilson tangles forms a U(1) group, which is a natural holonomy group of the CSW theory." @default.
- W73816557 created "2016-06-24" @default.
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- W73816557 date "1990-01-01" @default.
- W73816557 modified "2023-09-27" @default.
- W73816557 title "Tangles, Links and Twisted Quantum Groups" @default.
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- W73816557 doi "https://doi.org/10.1007/978-1-4615-3802-8_20" @default.
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