Quadruply-graded colored homology of knots

Gorsky, Eugene; Gukov, Sergei; Stošić, Marko

doi:10.48550/arxiv.1304.3481

Cited by 44 publications

(164 citation statements)

References 0 publications

Supporting

Mentioning

152

Contrasting

Unclassified

Order By: Relevance

“…Define a section ϕ m|n of B ∨ n with the value X m Y n v ∈ V at the fiber at the point (X, Y, v). Contraction with this section i ϕ m|n : Λ The differential d m|n is very much in spirit of the conjectures in the papers [35,13]…”

Section: Indeed Only Coordinatesmentioning

confidence: 94%

“…Hence H = π * 12 (F ) ⊗ π * 23 (G) and both " H and Φ (13) (H) provide an B 2 -equivariant extension of π * y (H) from X 3 to X 3 . Hence the result follows from the uniqueness of the extension.…”

mentioning

confidence: 98%

“…The later identity holds because the map π13 × π13 : X 3 × X 3 → X 2 × X 2 sends the kernel of Φ (13) , which is jx × πy ( X 3 ) ⊂ X 3 × X 3 to the kernel of the map Φ, which is j…”

mentioning

confidence: 99%

See 2 more Smart Citations

Knot homology and sheaves on the Hilbert scheme of points on the plane

Oblomkov

Rozansky

2018

Sel. Math. New Ser.

175

View full text Add to dashboard Cite

For each braid β ∈ Brn we construct a 2-periodic complex S β of quasi-coherent C * × C *equivariant sheaves on the non-commutative nested Hilbert scheme Hilb f ree 1,n . We show that the triply graded vector space of the hypercohomology H(S β ⊗ ∧ • (B)) with B being tautological vector bundle, is an isotopy invariant of the knot obtained by the closure of β. We also show that the support of cohomology of the complex S β is supported on the ordinary nested Hilbert scheme Hilb 1,n ⊂ Hilb f ree 1,n , that allows us to relate the triply graded knot homology to the sheaves on Hilb 1,n List of Symbols 81References 82

show abstract

Section: Indeed Only Coordinatesmentioning

confidence: 94%

mentioning

confidence: 98%

See 1 more Smart Citation

Knot homology and sheaves on the Hilbert scheme of points on the plane

Oblomkov

Rozansky

2018

Sel. Math. New Ser.

175

View full text Add to dashboard Cite

show abstract

“…Surprisingly enough, we prove that the Poincaré polynomials of the Heegaard-Floer homology of T (3, m) can be written using a formula with a similar structure. The corresponding differential corresponds to d 0 in the notations of [7,11] and to d 1|1 in the notations of [8].…”

Section: Introductionmentioning

confidence: 99%

On Stable -Homology of Torus Knots

Gorsky

Lewark

2015

Experimental Mathematics

Self Cite

View full text Add to dashboard Cite

The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for sl 3 -homology. Special attention is paid to torsion. In addition, explicit conjectural formulae are given for the sl N -homology of (3, m)-torus knots for all N and m, which are motivated by higher categorified Jones-Wenzl projectors. Structurally similar formulae are proven for Heegard-Floer homology.

show abstract

“…It generalizes the colored Jones, Alexander and the sl(N ) Reshetikhin-Turaev invariants. Their common property is that they arise from representations of sl(N |M ) superalgebras [10,11].…”

Section: Introductionmentioning

confidence: 99%

A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras

Mishnyakov,

Sleptsov,

Tselousov

2020

Preprint

View full text Add to dashboard Cite

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of N and generalizes the previously known "tug-the-hook" symmetry of the colored Alexander polynomial [1]. As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property imposes on the general group-theoretical structure, namely the sl(N ) weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture.

show abstract

Quadruply-graded colored homology of knots

Cited by 44 publications

References 0 publications

Knot homology and sheaves on the Hilbert scheme of points on the plane

Knot homology and sheaves on the Hilbert scheme of points on the plane

On Stable -Homology of Torus Knots

A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras

Contact Info

Product

Resources

About