This paper proves that certain monotone Lagrangians in the standard symplectic vector space cannot be displaced by a Hamiltonian isotopy which commutes with the antipodal map. The method of proof is to develop a Borel equivariant version of the quantum cohomology of Biran and Cornea and prove that it is sensitive to equivariant displacements. The Floer–Euler class of Biran and Khanevsky appears as a term in the equivariant differential in certain cases.
We present a new solution to the formality problem for the framed Goldman–Turaev Lie bialgebra, constructing Goldman-Turaev homomorphic expansions (formality isomorphisms) from the Kontsevich integral. Our proof uses a three dimensional derivation of the Goldman-Turaev Lie biaglebra arising from a low-degree Vassiliev quotient – the emergent quotient – of tangles in a thickened punctured disk, modulo a Conway skein relation. This is in contrast to Massuyeau's 2018 proof using braids. A feature of our approach is a general conceptual framework which is applied to prove the compatibility of the homomorphic expansion with both the Goldman bracket and the technically challenging Turaev cobracket.
For each braided category \mathcal{C} , we show that, under mild hypotheses, there is an associated category of “half braided algebras” and their bimodules internal to \mathcal{C} which is not only monoidal but even braided and balanced. We use this in the case where \mathcal{C} is the category of modules over a ribbon Hopf algebra to interpret stated skeins as a TQFT, namely, a braided balanced functor from a category of cobordisms to this category of algebras and their bimodules. Although our construction works in full generality, we relate in the special case of finite-dimensional ribbon factorizable Hopf algebras the stated skein functor to the Kerler–Lyubashenko TQFT by interpreting the former as the “endomorphisms” of the latter.
Motivated by the conjectured asymptotics of the Kashaev invariant, Dimofte and the first author introduced a power series associated to an essential ideal triangulation of a cusped hyperbolic 3-manifold, proved that its constant (1-loop) term is a topological invariant, and conjectured that it equals the adjoint Reidemeister torsion. We prove this conjecture for hyperbolic 2-bridge knots by combining the work of Ohtsuki-Takata with an explicit computation.
We prove that the Fukaya-Seidel categories of a certain family of singularities on $\mathbb{C}^d$ are equivalent to the perfect derived categories of higher Auslander algebras of Dynkin type A. We relate these to the Fukaya-Seidel categories of Brieskorn-Pham singularities and to the partially wrapped Fukaya categories considered by Dyckerhoff-Jasso-Lekili. We provide a symplectic interpretation to higher Auslander correspondence of type A in terms of Fukaya-Seidel categories of Lefschetz fibrations.
The Links-Gould invariant of links $LG^{2,1}$ is a two-variable generalization of the Alexander-Conway polynomial. Using representation theory of $U_{q}\mathfrak{gl}(2 \vert 1)$, we prove that the degree of the Links-Gould polynomial provides a lower bound on the Seifert genus of any knot, therefore improving the bound known as the Seifert inequality in the case of the Alexander invariant. One practical consequence of this new genus bound is a straightforward proof of the fact that the Kinoshita-Terasaka and Conway knots have genus greater or equal to 2.
We show that a contact (+1)-surgery along a Legendrian sphere in a flexibly fillable contact manifold ( c(1) = 0 if not subcritical) yields a contact manifold that is algebraically over-twisted if the Legendrian's homology class is not annihilated in the filling. Our construction can also be implemented in more general contact manifolds yielding algebraically overtwisted man-ifolds through (+1)-surgeries. This gives a new proof of the vanishing of contact homology for overtwisted contact manifolds. Our result can be viewed as the symplectic field theory analog in any dimension of the vanishing of contact Ozsv & aacute;th-Szab & oacute; invariant for (+1)-surgeries on two-component Legendrian links proved by Ding, Li, and Wu (2020).
We consider the link and three-manifold invariants in [DGGPR], which are defined in terms of certain non-semisimple finite ribbon categories $\mathcal{C}$ together with a choice of tensor ideal and modified trace. If the ideal is all of $\mathcal{C}$, these invariants agree with those defined by Lyubashenko in the 90's. We show that in that case the invariants depend on the objects labelling the link only through their simple composition factors, so that in order to detect non-trivial extensions one needs to pass to proper ideals. We compute examples of link and three-manifold invariants for $\mathcal{C}$ being the category of $N$ pairs of symplectic fermions. Using a quasi-Hopf algebra realisation of $\mathcal{C}$, we find that the Lyubashenko-invariant of a lens space is equal to the order of its first homology group to the power $N$, a relation we conjecture to hold for all rational homology spheres. For $N \ge 2$, $\mathcal{C}$ allows for tensor ideals $\mathcal{I}$ with a modified trace which are different from all of $\mathcal{C}$ and from the projective ideal. Using the theory of pull-back traces and symmetrised cointegrals, we show that the link invariant obtained from $\mathcal{I}$ can distinguish a continuum of indecomposable but reducible objects which all have the same composition series.
We prove an unoriented skein extract triangle for the real monopole Floer homology and introduce a Frøyshov-type invariant.
We show that cellular Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The chain-level map replaces the two orderings of the self-intersection point with meridianal and longitudinal cells on the handle created by the surgery, and uses a bijection between holomorphic disks developed by Fukaya-Oh-Ohta-Ono. Our result generalizes invariance of potentials for certain Lagrangian surfaces in Dimitroglou-Rizell--Ekholm--Tonkonog, and implies the invariance of Floer cohomology under mean curvature flow with this type of surgery, as conjectured by Joyce.
If $C$ is a spherical fusion category, the string-net construction associates to each closed oriented surface $\Sigma$ the vector space $Z_\text{SN}(\Sigma)$ of linear combinations of $C$-labelled graphs on $\Sigma$ modulo local relations, in a way which is functorial with respect to orientation-preserving diffeomorphisms of surfaces. We show how to extend this assignment to a 3-dimensional topological quantum field theory (TQFT), by defining how the surgery generators in Juhász' presentation of the oriented 3-dimensional bordism category act on the string-net vector spaces. We show that the resulting TQFT, which is formulated completely in the two-dimensional graphical language of string-nets, is an alternative description of the Turaev-Viro state sum model.
Let L be a null homologous link in $\mathbb{RP}^3$. We define Khovanov-type homologies of L which depend on an extra input $\alpha = (V_0,V_1,f,g)$ consisting of two graded vectors spaces and two maps between them. With some specific choice of $\alpha = \alpha_{APS}$, we recover the categorification of the Kauffman bracket due to Asaeda-Przytycki-Sikora. With another choice of $\alpha = \alpha_{HF}$, we construct a spectral sequence from our theory converging to the Heegaard Floer homology of the even branched double cover of $\mathbb{RP}^3$.
We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra g is a Noetherian and finitely generated ring. If the surface has punctures, we also prove that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin-Turaev functor for the quantum group U-q(g), and which coincides with the Kauffman bracket skein algebra when g = sl(2). We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.
A unitary fusion category is called Z/2Z-quadratic if it has a Z/2Z group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of Z/2Z-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalisation of Ostrik's results on formal codegrees to analyse the induction of the group elements to the centre, and a computation similar to Larson's rank-finiteness bound for Z/3Z-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information.
We show that the ribbon zesting construction can produce modular isotopes —different modular fusion categories with the same modular data. The result relies on the observation that the Reshetikhin–Turaev invariants of framed links associated to a ribbon fusion category satisfy a factorization property under zesting. This gives a new perspective on using topological invariants to classify topological order in light of modular data not being a complete invariant.
We prove the conjecture of Przytycki and Sazdanović that the Khovanov homology of the closure of a 3-stranded braid only contains torsion of order 2. This conjecture has been known for six out of seven classes in the Murasugi-classification of 3-braids, and we show it for the remaining class. Our proof also works for the other classes and relies on Bar-Natan’s version of Khovanov homology for tangles as well as his delooping and cancellation techniques and the reduced integral Bar-Natan–Lee–Turner spectral sequence. We also show that the Knight move conjecture holds for 3-braids.
In this note, we construct dual PBW bases of the positive and negative subalgebras of the two-parameter quantum groups U-r,U-s(g) in classical types, as used in "Martin-Tsymbaliuk [SIGMA 21 (2025), paper no. 064]". Following the ideas of "Leclerc [Math. Z. 246 (2004), no. 4, 691-732]" and "Clark-Hill-Wang [Quantum Topol. 7 (2016), no. 3, 553-638]", we introduce the two-parameter shuffle algebra and relate it to the subalgebras above. We then use the combinatorics of dominant Lyndon words to establish the main results.
We present a state sum construction that assigns a scalar to a skeleton in a closed oriented three-dimensional manifold. The input datum is the pivotal bicategory \mathbf{Mod}^{\mathrm{sph}}(\mathcal{A}) of spherical module categories over a spherical fusion category \mathcal{A} . The interplay of algebraic structures in this pivotal bicategory with moves of skeleta ensures that our state sum is independent of the skeleton on the manifold. We show that the bicategorical invariant recovers the value of the standard Turaev–Viro invariant associated to \mathcal{A} , thereby proving the independence of the Turaev–Viro invariant under pivotal Morita equivalence without recurring to the Reshetikhin–Turaev construction. A key ingredient for the construction is the evaluation of graphs on the sphere with labels in \mathbf{Mod}^{\mathrm{sph}}(\mathcal{A}) that we develop in this article. A central tool is Nakayama-twisted traces on pivotal bimodule categories, which we study beyond semisimplicity.
We define a new algebra associated to a Legendrian submanifold \Lambda of a contact manifold of the form \mathbb{R}_{t}\times \mathrm{W} , called the planar diagram algebra , and denote it by \mathrm{PDA}(\Lambda, \mathcal{P}) . It is a non-commutative, filtered, differential graded algebra whose filtered stable tame isomorphism class is an invariant of \Lambda together with a partition \mathcal{P} of its connected components. When \Lambda is connected, \mathrm{PDA} is the Chekanov–Eliashberg algebra. In general, the \mathrm{PDA} differential counts holomorphic disks with multiple positive punctures using a combinatorial framework inspired by string topology.
This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the $E_0$-center ($E_1$-center or $E_2$-center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series.
