Related papers: $\Delta$ Invariants of Plumbed Manifolds
An invariant is introduced for negative definite plumbed $3$-manifolds equipped with a spin$^c$-structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the…
Recently a set of $q$-series invariants, labelled by $\operatorname{Spin}^c$ structures, for weakly negative definite plumbed $3$-manifolds called the $\widehat{Z}_a$ invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading…
We consider a connected negative definite plumbing graph, and we assume that the associated plumbed 3-manifold is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg-Witten invariant of this manifold. The…
We study equivariant Seiberg-Witten Floer theory of rational homology $3$-spheres in the special case where the group action is given by an involution. The case of involutions deserves special attention because we can couple the involution…
The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsvath and Szabo for a family of plumbed rational homology 3-spheres. The main motivation is the study of the Seiberg-Witten type…
We study formal series which are invariants of plumbed 3-manifolds twisted by root lattices. These series extend the BPS $q$-series $\widehat{Z}(q)$ recently defined in Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and further refined in…
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…
Let M be a rational homology sphere plumbed 3-manifold associated with a connected negative definite plumbing graph. We show that its Seiberg-Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial.…
We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic $q$. Both types of invariants are labeled by extra data which plays an important role in the…
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in [arXiv:1404.7289]. In their construction the quantum parameter $q$ is a root of unity of order…
We introduce supergroup analogues of 3-manifold invariants $\hat{Z}$, also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups, and work out the case of…
We introduce a two-variable refinement $\hat{Z}_a(q,t)$ of plumbed 3-manifold invariants $\hat{Z}_a(q)$, which were previously defined for weakly negative definite plumbed 3-manifolds. We also provide a number of explicit examples in which…
Gukov, Pei, Putrov, and Vafa developed a $q$-series invariant of negative definite plumbed $3$-manifolds with spin$^{c}$ structures, building on earlier work of Lawrence and Zagier. This was recently generalized to an an infinite family of…
We study the quantum modular properties of $\widehat Z{}^G$-invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups $G$. In particular, we…
We provide a physical definition of new homological invariants $\mathcal{H}_a (M_3)$ of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on $M_3$ times a…
In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a ${\mathbb Q}$-Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider…
In this paper we study an invariant for oriented three-manifolds with $b_1>0$, which is defined using Heegaard splittings and the theta divisor of a Riemann surface. The paper is divided into two parts, the first of which gives the…
We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they…
We introduce an invariant of negative definite plumbed knot complements unifying knot lattice homology, due to Ozsv\'ath, Stipsicz, and Szab\'o, and the BPS $q$-series of Gukov and Manolescu. This invariant is a natural extension of…
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer…