arXiv research

Identifies 3D-index as invariant for cusped hyperbolic 3-manifolds.

New indefinite false theta functions match homological blocks for a specific 3-manifold.

We study $q$-series-valued invariants of 3-manifolds that depend on the choice of a root system $G$. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057] where they focused on $G={\rm SU}(2)$ case. Although a full mathematical definitio…

Proves a quantum invariant conjecture for specific three-manifolds.

Study series invariants for plumbed 3-manifolds and their properties.

Study series invariants of plumbed 3-manifolds using root lattices.

New formula and properties of inverted Habiro series derived from GM series.

In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL_2(C) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the corresponding $\mathbb{C}^{*}$-valued closed 1-form is a secondary characteristic clas…

New knot homology invariant grows exponentially with color.

Proves a conjecture about 3-manifold invariants using trees and asymptotic formulas.

We propose an algebraic model of the conjectural triply graded homology of Gukov, Dunfield and Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of Garsia and Haiman.

Researchers calculate $(t,q)$-series invariants for Seifert manifolds.

fkcompute calculates a knot invariant from a braid presentation.

New formulas connect knot invariants with theta functions.

This paper studies the formal adiabatic limit of coassociative K3 fibred torsion free $G_2$ manifolds fibred over a contractible base, shows how to put this structure on a different fibration obtained by fibrewise performing Mukai duality of K3 surfaces, and furthermore relates the gauge theories on both fibrations by …

New operators in Khovanov-Rozansky homology exhibit symmetry.

Mathematical framework for brane quantization using SYZ mirror symmetry.

This paper proves a conjecture about knot homologies.

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special linear group of degree two over complex numbers. We also confirm the conjecture for…

We compute the asymptotical growth rate of a large family of $U_q(sl_2)$ $6j$-symbols and we interpret our results in geometric terms by relating them to volumes of hyperbolic truncated tetrahedra. We address a question which is strictly related with S.Gukov's generalized volume conjecture and deals with the case of hy…

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 phys…

Decorated TQFTs compute invariants with additional structures.

In this paper we study new invariants $\widehat{Z}_{\boldsymbol{a}}(q)$ attached to plumbed $3$-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable $q$-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed $3$-manifold. Here we investigate the series $\wi…

In fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2 theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomia…

Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating serie…

The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposa…

New invariant connects knot homology and BPS series for plumbed knot complements.

Researchers found the $\hat{Z}$-invariant for $SU(N)/\mathbb{Z}_m$ is constant regardless of $m$.

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 algebra…

Researchers create a new invariant for knot theory.

The paper defines and computes a knot complement invariant for simple links.

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index…

Defines a map connecting 3d-index and skein module.

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there…

For a Seifert fibered homology sphere we show that the q-series Z-hat invariant introduced by Gukov, Pei, Putrov and Vafa is a resummation of the Ohtsuki serie. We show that for every even level k there exists a full asymptotic expansion of Z-hat for q tending to a certain k'th root of unity and in particular that the …

The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with $r$ torii boundary components. For a fixed $2r$ tuple of integers, the index takes values in the set of $q$-series with integer coefficients. Our goal is to give an axiomatic definition of the tetra…

The tetrahedral index connects to a q-Bessel function, revealing new mathematical techniques.

Study knot invariants to deduce Hopf invariant and propose a slope conjecture.

Geometric quantization on hyperKähler manifolds via brane quantization.

The paper connects knot homology with sheaf theory and proves symmetry properties.

The paper connects quivers to knot complements and studies their BPS states and 3d N=2 theories.

The abstract discusses resurgent functions in quantum knot invariants.

Study of 3d-3d correspondence involving $q$-Weyl algebra and 3d-index.

This paper analyzes $Δ_a$ invariants in non-perturbative complex Chern-Simons theory.

In this paper, by using the regulator map of Beilinson-Deligne, we show that the quantization condition posed by Gukov is true for the SL_2(\mathbb{C}) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the corresponding \mathbb{C}^{*}-valued 1-form is a secondary characteristic class (Chern-Si…

Innovative series invariant for knot complements, linking to existing invariants.

Extends knot invariant computation to symmetrically colored sl_N.