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The universal Khovanov link homology theory
Gad Naot
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Algebraic & Geometric Topology 6
(2006) 1863–1892
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Abstract
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We determine the algebraic structure underlying the geometric complex
associated to a link in Bar-Natan's geometric formalism of Khovanov's
link homology theory (n=2). We find an isomorphism of complexes
which reduces the complex to one in a simpler category. This
reduction enables us to specify exactly the amount of information held
within the geometric complex and thus state precisely its universality
properties for link homology theories. We also determine its strength
as a link invariant relative to the different topological quantum
field theories (TQFTs) used to create link homology. We identify the
most general (universal) TQFT that can be used to create link homology
and find that it is “smaller” than the TQFT previously reported by
Khovanov as the universal link homology theory. We give a new method
of extracting all other link homology theories (including Khovanov's
universal TQFT) directly from the universal geometric complex, along
with new homology theories that hold a controlled amount of
information. We achieve these goals by making a classification of
surfaces (with boundaries) modulo the 4TU/S/T relations, a process
involving the introduction of genus generating operators. These
operators enable us to explore the relation between the geometric
complex and its algebraic structure.
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Keywords
categorification, cobordism, Jones
polynomial, Khovanov link homology, quantum knot invariants,
TQFT
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Mathematical Subject Classification
Primary: 57M25
Secondary: 57M27
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Publication
Received: 19 May 2006
Revised: 14 July 2006
Accepted: 20 July 2006
Published: 1 November 2006
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