In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the B-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and, as an application, we prove derived equivalences between certain nodal stacky curves, some of whose irreducible components have non-trivial generic stabiliser

We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $\mathfrak{sl_2}$ using purely combinatorial methods based on Temperley–Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten–Reshetikhin–Turaev invariants, which can be reformulated following our approach

For a given smooth 2-knot in the 4-space, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible SU(2)-representations of its knot group. For example, we see that any smooth 2-knot having the Poincar ́e homology 3-sphere as a Seifert hypersurface has at least four irreducible SU(2)-representations of its knot group. This result can not be proved

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z\colon \mathcal{B}\to \hat{A}$, where $\mathcal{B}$ is the category of bottom tangles in handlebodies and $\hat{A}$ is the degree-completion of the category $\mathbf{A}$ of Jacobi diagrams in handlebodies. As a symmetric monoidal linear

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of $\mathfrak{sl}_2$. From the above we

We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $\mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g \geq 1$ with $n \geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded

In this paper we introduce an extension of the hat Heegaard Floer TQFT which allows cobordisms with disconnected ends. Our construction goes by way of sutured Floer homology, and uses some elementary results from contact geometry. We provide some model computations, which allow us to realize the $H_1(Y;\mathbb{Z})/\text{Tors}$ action and the first order term, $\partial_1$, of the differential of $CF^\infty$

In previous work, we introduced a natural $\mathcal{A}_{\infty}$-structure on the Pin(2)-monopole Floer chain complex of a closed, oriented three-manifold $Y$, and showed that it is non-formal in the simplest case in which $Y$ is the three-sphere $S^3$. In this paper, we explore further this non-formality phenomenon. Specifically, we provide explicit descriptions of several Massey products induced

We define a faithful linear monoidal functor from the partition category, and hence from Deligne’s category $\mathrm{\underline{Re}p}(S_t)$, to the additive Karoubi envelope of the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions.

For any positive integer $k$ and $p\in \{3,5,7\}$ we construct a link which has a direct summand $\mathbb Z/p^k\mathbb Z$ in its Khovanov cohomology.

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmüller space. In this paper

Let a contact 3-manifold $(Y, \xi_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $\xi_0$. We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant $c^+(\xi_0)\in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. It follows that Karakurt's "height of $U$-tower" invariants are always 0 for

We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.

The physical 3d $\mathcal N = 2$ theory $T[Y]$ was previously used to predict the existence of some $3$-manifold invariants $\widehat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten–Reshetikhin–Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the

In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author in [4]. The $\Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern–Simons perturbation theory. The $\Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding $\mathfrak {gl}_2$ skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of $\mathfrak {gl}_2$ foams that admits an interesting

In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, which we call IBL$_\infty$-algebras.

We prove the unrolled superalgebra $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $\xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.

Let $\Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $\mathfrak{g}(\Sigma)$ and its associated graded ${\rm gr}\, \mathfrak{g}(\Sigma)$

We give a purely combinatorial formula for evaluating closed, decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral, equivariant version of colored Khovanov–Rozansky link homology categorifying the $\mathfrak {sl}_N$ link polynomial. We also provide connections to the equivariant cohomology rings of partial flag varieties.

The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that twisted, generalized Whitehead

Let $\mathcal{C}(\mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $\mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $\mathcal{C}(\mathfrak{g},k)_R^0$ where $R$

This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics

In the late 1980's, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration. This was interpreted as the secondary characteristic class $d_1$ associated with the fact that the first MMM class vanishes on the Torelli group. It is a rational generator of $H^1(\mathcal{K}_g;\mathbb{Z})^{\mat

We equip Ellis and Brundan's version of the odd categorified quantum group for sl(2) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible decategorifications of the associated dg-2-category. One version gives rise to a categorification of quantum sl(2) at a fourth root of unity, while the other version produces a subalgebra

In this paper, we prove that the Ozsv\'ath-Szab\'o invariant of a contact 3-manifold vanishes if it can be obtained by contact $+1$ surgery on the standard contact 3-sphere along a Legendrian two-component link whose linking number is nonzero and the topological type of one of whose components is smoothly slice. As a corollary, the Ozsv\'ath-Szab\'o invariant of a contact $\frac{1}{n}$ surgery on the

We give a formula for the Heegaard Floer $d$-invariants of integral surgeries on two-component L--space links of linking number zero in terms of the $h$-function, generalizing a formula of Ni and Wu. As a consequence, we characterize L-space surgery slopes for such links in terms of the $\tau$-invariant when the components are unknotted. For general links of linking number zero, we explicitly describe

In a previous paper, V\'ertesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by gluing together $T_1, \dotsc, T_m$, then the knot Floer homology $\widehat{\mathrm{HFK}}(L)$ of $L$ can be recovered from $\widetilde{\mathrm{CT}} (T_1), \dotsc, \wi

We establish relationships between two classes of invariants of Legendrian knots in $\mathbb{R}^3$: Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, $\beta \subset J^1S^1$, we give a precise formula in terms of representation numbers for the $m$-graded ruling polynomial $R^m_{S(K,\beta)}(z)$ of the satellite of $K$ with $\beta$

Let $L \subset \mathbb R \times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends $\Lambda_\pm \subset J^1(M)$. It is well known that the Legendrian contact homology of $\Lambda_\pm$ can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent

Given an arbitrary graph $\Gamma$ and non-negative integers $g_v$ for each vertex $v$ of $\Gamma$, let $X_\Gamma$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*\Sigma_v$ according to this graph, where $\Sigma_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_\Gamma$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922

We construct a triangulated monoidal Karoubi closed category with the Grothendieck ring, naturally isomorphic to the ring of integers localized at two.

Author(s): Cui, Xingshan | Advisor(s): Wang, Zhenghan | Abstract: We give a construction of Turaev-Viro type (3+1)-TQFT out of a G-crossed braided spherical fusion category for G a finite group. The resulting invariant of 4-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant and Yetter's invariant from homotopy 2-types. Some concrete examples will be provided

We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometric interpretation of the maps appearing in the long exact sequence for symplectic

This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.

Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either $E^6$ or the ones from representations

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $\rho$ of the skein algebra, which is a character $r_\rho \in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ represented by a group homomorphism

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $\Upsilon_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, closed surface in $S^4$ are determined by the genus of the surface. We also

We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the Borromean rings complement

We relate decategorifications of Ozsv\'ath-Szab\'o's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsv\'ath- Szab\'o's algebra $\mathcal{B}(n,\mathcal{S})$, and identify their Grothendieck groups with tensor products of representations

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $\Delta$-separable symmetric Frobenius algebras

Let $\mathcal{L}$ be a knot with a fixed positive crossing and $\mathcal{L}_n$ the link obtained by replacing this crossing with $n$ positive twists. We prove that the knot Floer homology $\widehat{\text{HFK}}(\mathcal{L}_n)$ `stabilizes' as $n$ goes to infinity. This categorifies a similar stabilization phenomenon of the Alexander polynomial. As an application, we construct an infinite family of prime

We categorify a coideal subalgebra of the quantum group of $\mathfrak{sl}_{2r+1}$ by introducing a $2$-category a la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the

We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra $\text{TL}_k(d)$, converging for all complex loop parameters $d$ with $|d| > 2\cos\big(\frac{\pi}{k+1}\big)$. In particular, this yields a new formula for the structure coefficients of the Jones-Wenzl projection in $\text{TL}_k(d)$

We prove that the Witten--Reshetikhin--Turaev $\mathrm{SU}(2)$ quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by one. We thus generalize a well--known result due to J. Roberts.

We investigate a (potentially infinite) series of subfactors, called $3^n$ subfactors, including $A_4$, $A_7$, and the Haagerup subfactor as the first three members corresponding to $n=1,2,3$. Generalizing our previous work for odd $n$, we further develop a Cuntz algebra method to construct $3^n$ subfactors and show that the classification of the $3^n$ subfactors and related fusion categories is reduced

By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general

We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $\exp({2\pi\sqrt{-1}}/{r})$ instead of the standard $\exp({\pi\sqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $\infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold

We show how to construct unitary representations of the oriented Thompson group $\vec{F}$ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $\vec{F}$.

We define a generalization of the idempotented Schur algebra which gives a unified setting for a quantum group presentation of the Homfly-Pt polynomial, together with its specializations to the Alexander polynomial and the slm Reshetikhin-Turaev invariants.

We give a presentation of the $n$-dimensional oriented cobordism category $\text{Cob}_n$ with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor $F$ from the category of smooth oriented manifolds and diffeomorphisms to an arbitrary category $C$, and morphisms induced by surgeries along framed spheres, we obtain a necessary

In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories defined by Colin-Ghiggini-Honda preserves this grading.

We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many computations of framed instanton homology in the presence of a non-trivial real