Refined knot invariants and Hilbert schemes
By E. Gorsky, Andrei Neguct. 2013
199 citations recorded by the source corpus.
We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on the plane. Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the m/n shuffle conjecture.
View Paper: https://www.semanticscholar.org/paper/c5d69cd075ec4843d3e8fba113b4bba67be4e177
How researchers see this paper
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- I. Cherednik
- Andrei Neguct
- Alexei Oblomkov
- I. Danilenko
- H. Morton
- E. Gorsky
- Jacob Rasmussen
- François Bergeron
Citations in the corpus
- Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology (2016, methodology): “The first (already present in [30]) is that one should use the flag Hilbert scheme rather than the usual Hilbert scheme.”
- THE HOMFLYPT SKEIN ALGEBRA OF THE TORUS AND THE ELLIPTIC HALL ALGEBRA (2014, methodology): “Our construction uses the work of Schiffmann and Vasserot [SV13,SV11] in an essential way, and when restricted to torus knots, it is essentially the same as the construction of Gorsky and Negut [GN13].”
- The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, background): “It would take another paper to properly discuss all of the new results and connections between them, so here we just list a few references which we find most relevant.”
- Compositional (km,kn)-Shuffle Conjectures (2014, background): “Negut, in a period of several months, made us aware of some of the contents of the latter publications as well as the results in their papers ([14], [19] and [20]) in a language we could understand.”
- Rational Parking Functions and Catalan Numbers (2014, background): “See [14, 15, 23] for related work that arises from a more geometric viewpoint.”
- DAHA and iterated torus knots (2014, methodology): “Concerning the stabilization and duality, we mainly follow here [22], switching from P to the modified Macdonald polynomials.”