30 research outputs found
Categorified sl(N) invariants of colored rational tangles
We use categorical skew Howe duality to find recursion rules that compute
categorified sl(N) invariants of rational tangles colored by exterior powers of
the standard representation. Further, we offer a geometric interpretation of
these rules which suggests a connection to Floer theory. Along the way we make
progress towards two conjectures about the colored HOMFLY homology of rational
links.Comment: 45 pages, many figures, uses dcpic.sty, v2: minor changes and new
example 5
Exponential growth of colored HOMFLY-PT homology
We define reduced colored sl(N) link homologies and use deformation spectral
sequences to characterize their dependence on color and rank. We then define
reduced colored HOMFLY-PT homologies and prove that they arise as large N
limits of sl(N) homologies. Together, these results allow proofs of many
aspects of the physically conjectured structure of the family of type A link
homologies. In particular, we verify a conjecture of Gorsky, Gukov and
Sto\v{s}i\'c about the growth of colored HOMFLY-PT homologies.Comment: 39 pages, many figures. v2: version accepted for publication in
Advances in Mathematic
Rational links and DT invariants of quivers
We prove that the generating functions for the colored HOMFLY-PT polynomials
of rational links are specializations of the generating functions of the
motivic Donaldson-Thomas invariants of appropriate quivers that we naturally
associate with these links. This shows that the conjectural links-quivers
correspondence of Kucharski-Reineke-Sto\v{s}i\'c-Su{\l}kowski as well as the
LMOV conjecture hold for rational links. Along the way, we extend the
links-quivers correspondence to tangles and, thus, explore elements of a skein
theory for motivic Donaldson-Thomas invariants.Comment: 25 pages, comments welcom
Evaluations of annular Khovanov--Rozansky homology
We describe the universal target of annular Khovanov-Rozansky link homology
functors as the homotopy category of a free symmetric monoidal category
generated by one object and one endomorphism. This categorifies the ring of
symmetric functions and admits categorical analogues of plethystic
transformations, which we use to characterize the annular invariants of Coxeter
braids. Further, we prove the existence of symmetric group actions on the
Khovanov-Rozansky invariants of cabled tangles and we introduce spectral
sequences that aid in computing the homologies of generalized Hopf links.
Finally, we conjecture a characterization of the horizontal traces of Rouquier
complexes of Coxeter braids in other types.Comment: 41 page
Functoriality of colored link homologies
We prove that the bigraded colored Khovanov-Rozansky type A link and tangle
invariants are functorial with respect to link and tangle cobordisms.Comment: 41 pages, many colored figures, some changes following suggestions of
a referee, to appear in Proc. Lond. Math. Soc., comments welcom
Deformations of colored sl(N) link homologies via foams
We generalize results of Lee, Gornik and Wu on the structure of deformed
colored sl(N) link homologies to the case of non-generic deformations. To this
end, we use foam technology to give a completely combinatorial construction of
Wu's deformed colored sl(N) link homologies. By studying the underlying
deformed higher representation theoretic structures and generalizing the
Karoubi envelope approach of Bar-Natan and Morrison we explicitly compute the
deformed invariants in terms of undeformed type A link homologies of lower rank
and color.Comment: 64 pages, many figure
Super -Howe duality and web categories
We use super -Howe duality to provide diagrammatic presentations of an
idempotented form of the Hecke algebra and of categories of
-modules (and, more generally, -modules)
whose objects are tensor generated by exterior and symmetric powers of the
vector representations. As an application, we give a representation theoretic
explanation and a diagrammatic version of a known symmetry of colored
HOMFLY--PT polynomials.Comment: 38 pages, many colored figures, extra section containing new results,
added suggestions of two referees, comments welcom
A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain
The hamiltonian of the -state superintegrable chiral Potts (SICP) model is
written in terms of a coupled algebra defined by types of Temperley-Lieb
generators. This generalises a previous result for obtained by J. F.
Fjelstad and T. M\r{a}nsson [J. Phys. A {\bf 45} (2012) 155208]. A pictorial
representation of a related coupled algebra is given for the case which
involves a generalisation of the pictorial presentation of the Temperley-Lieb
algebra to include a pole around which loops can become entangled. For the two
known representations of this algebra, the SICP chain and the staggered
spin-1/2 XX chain, closed (contractible) loops have weight and
weight , respectively. For both representations closed (non-contractible)
loops around the pole have weight zero. The pictorial representation provides a
graphical interpretation of the algebraic relations. A key ingredient in the
resolution of diagrams is a crossing relation for loops encircling a pole which
involves the parameter for the SICP chain and
for the staggered XX chain. These values are derived assuming
the Kauffman bracket skein relation.Comment: 10 pages, 4 figures, further cubic relations adde