Categorifications of the colored Jones polynomial

By Mikhail Khovanov. 2003

145 citations recorded by the source corpus.

The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a81e7ee2d3e8f9ccc03c624d63de05d398cc7ebe

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Categorification of the Jones-Wenzl Projectors (2010, methodology): “It is important to note that our definition disagrees with some previous categorifications based on different axiomatizations of the Jones–Wenzl projectors such as the dimension axiom [17] (for related work see [5] and [11]): Tr.”
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links (2005, methodology): “Our first approach is inspired by Khovanov [6].”
BPS spectra and 3-manifold invariants (2017, background): “Although this is no longer the case in homological world, there is still a close relation between “cabling” and “color” [80,81].”
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology (2008, background): “In [12], Khovanov associates to a knot, K ⊂ S(3), a bigraded homology group for each n ∈ Z>0, here denoted K̃hn(K), whose graded Euler characteristic is the reduced n–colored Jones polynomial, J̃n(K), of K:”
sl(3) link homology (2003, unknown): “Its categorification is sketched in [4].”
Khovanov-Rozansky Homology and Topological Strings (2004, methodology): “In another line of development, the quantum sl(N) invariant PN (q) was lifted to a homological knot invariant [14,15,16,17].”
Fixing the functoriality of Khovanov homology (2007, methodology): “Remark Something like this lemma is used in [15] in describing a categorification of the colored Jones polynomial, without the need for functoriality.”
Annular Khovanov homology and knotted Schur–Weyl representations (2015, methodology): “Then the subspace of Sn invariants inside the Khovanov homology of K (with coefficients in a field of characteristic 0) is isomorphic to Khovanov’s categorification of the non-reduced n-colored Jones polynomial of K [23].”
Functoriality of colored link homologies (2017, methodology): “One uses finite resolutions of these representations by fundamental representations and is analogous to Khovanov’s construction for N = 2 in [24], see also Robert [44] for the case N = 3.”
Categorifying fractional Euler characteristics, Jones–Wenzl projectors and 3j-symbols (2012, background): “On the other hand, our approach gives rise to a different (and actually much more subtle) colored Jones invariant than the original construction of Khovanov in [36] and the one in [13].”