Torus knots and the rational DAHA
By E. Gorsky, Alexei Oblomkov, Jacob Rasmussen, Vivek V. Shende. 2012
141 citations recorded by the source corpus.
Author(s): Gorsky, E; Oblomkov, A; Rasmussen, J; Shende, V | Abstract: © 2014. We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m;n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q; t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1c6f366519f391aebb197737b488d6e831eec4f1
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In Geometry and Topology
- E. Gorsky
- Andrei Neguct
- I. Danilenko
- Jacob Rasmussen
- I. Cherednik
- Alexei Oblomkov
- Vivek V. Shende
- I. Losev
In Mathematical Physics
- E. Gorsky
- I. Cherednik
- Alexei Oblomkov
- Andrei Neguct
- Vivek V. Shende
- Jacob Rasmussen
- I. Danilenko
- I. Losev
Citations in the corpus
- Refined knot invariants and Hilbert schemes (2013, methodology): “cture 5.4 is also supported by numerical computations, and it is compatible with some structural properties. For example, the symmetry between the q and t gradings of the character has been proven in [31], and this symmetry is manifest in the operators Pe n,m. Moreover, Conjecture 5.4 was proved in ”
- Affine permutations and rational slope parking functions (2014, background): “admits an affine paving by m n−1 affine cells. It turns out that the affine paving of this affine Springer fiber is tightly related to the combinatorics of the simplex Dm n . This was implicitly stated in [17, 20, 24], but we would like to make this correspondence precise and explicit. 26 EUGENE GORSKY, ”
- Representations of Rational Cherednik algebras with minimal support and torus knots (2013, methodology): “Our motivation came from the connections of representations of Cherednik algebras with quantum invariants of torus knots and Hilbert schemes of plane curve singularities (such as x = y, GCD(m,n) = 1), see [GORS].”
- The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, methodology): “We will explore this idea more fully in a subsequent paper [34].”
- Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology (2016, background): “This description of dN indeed agrees with the ones in [29, 31, 32], and the homology is quite involved.”
- DAHA and iterated torus knots (2014, methodology): “A similar connection is also expected with the physics superpolynomials associated with the BPS states (see eg [15; 1; 16]) and those related to the Hilbert schemes of plane curve singularities and rational DAHA; see [33; 23; 22] and references therein.”
- Toric braids and (m,n)-parking functions (2016, result): “s by raising and lowering operators. Existence of these operators is very useful for the present paper. No attempt was made to compare techniques of the present work with previous works on DAHA, i.e. [GORS14] and [CD14]. We can expect that such a comparison would produce many interesting results. I ”