Abstract We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise nonisotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive answer to a stronger version of an old question of Hitt and Sumners. The examples arise from our main result: a classification of fibered, homotopy‐ribbon disks for each generalized square knot up to isotopy. Precisely, we show that each generalized square knot bounds infinitely many pairwise nonisotopic fibered, homotopy‐ribbon disks, all of whose exteriors are diffeomorphic. When , we prove further that infinitely many of these disks are also ribbon; whether the disks are always ribbon is an open problem.

Abstract The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analog for an isolated singularity. We define the monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class in the symplectic cohomology of the inverse of the monodromy and its closed–open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of noncompact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator.

Abstract We prove the existence of a subclass of overtwisted contact structures, called strongly overtwisted, on a 3‐manifold that satisfy a complete ‐principle without prescribing the contact structures over any subset of the 3‐manifold. As a consequence, the homotopy type of the space of overtwisted disk embeddings into a strongly overtwisted contact 3‐manifold is determined. A complete ‐principle for a subclass of loose Legendrians is also derived from the main result. In general, the method allows us to deduce an ‐principle for overtwisted disks that are fixed near the boundary in an arbitrary overtwisted contact 3‐manifold.

Abstract We develop a framework of parametrized semiadditivity and stability with respect to so‐called atomic orbital subcategories of an indexing ‐category , extending work of Nardin. Specializing this framework, we introduce global ‐categories and the notions of equivariant semiadditivity and stability, yielding a higher categorical version of the notion of a Mackey 2‐functor studied by Balmer–Dell'Ambrogio. As our main result, we identify the free presentable equivariantly stable global ‐category with a natural global ‐category of global spectra for finite groups, in the sense of Schwede and Hausmann.

Abstract We refine several results of Bhatt–Morrow–Scholze on to real topological Hochschild homology (). In particular, we compute of perfectoid rings. This will be useful for establishing motivic filtrations on real topological Hochschild and cyclic homology of quasisyntomic rings. We also establish a real refinement of the Hochschild–Kostant–Rosenberg theorem.

Abstract We establish the geometry behind the quantum ‐symbols under only the admissibility conditions as in the definition of the Turaev–Viro invariants of 3‐manifolds. As a classification, we show that the 6‐tuples in the quantum ‐symbols give in a precise way to the dihedral angles of (1) a spherical tetrahedron, (2) a generalized Euclidean tetrahedron, (3) a generalized hyperbolic tetrahedron or (4) in the degenerate case the angles between four oriented straight lines in the Euclidean plane. We also show that for a large proportion of the cases, the 6‐tuples always give the dihedral angles of a generalized hyperbolic tetrahedron and the exponential growth rate of the corresponding quantum ‐symbols equals the suitably defined volume of this generalized hyperbolic tetrahedron. It is worth mentioning that the volume of a generalized hyperbolic tetrahedron can be negative, hence the corresponding sequence of the quantum ‐symbols could decay exponentially. This is a phenomenon that has never been seen before.

Abstract For any Legendrian knot or link in , we construct an algebra that can be viewed as an extension of the Chekanov–Eliashberg differential graded algebra. The structure incorporates information from rational symplectic field theory and can be formulated combinatorially. One consequence is the construction of a Poisson bracket on commutative Legendrian contact homology, and we show that the resulting Poisson algebra is an invariant of Legendrian links under isotopy.

Abstract We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. The limit sets under consideration are defined in a general convergence compactification, including Gromov boundary, Bowditch boundary, Thurston boundary and horofunction boundary. As an application, we confirm a conjecture of Maher that hyperbolic 3‐manifolds are exponentially generic in the set of 3‐manifolds built from Heegaard splitting using complexity in Teichmüller metric.

Abstract We calculate the Dehn twist action on the spaces of conformal blocks of a not necessarily semisimple modular category. In particular, we give the order of the Dehn twists under the mapping class group representations of closed surfaces. For Dehn twists about non‐separating simple closed curves, we prove that this order is the order of the ribbon twist, thereby generalizing a result that De Renzi–Gainutdinov–Geer–Patureau–Mirand–Runkel obtained for the small quantum group. In the separating case, we express the order using the order of the ribbon twist on monoidal powers of the canonical end. As an application, we prove that the Johnson kernels of the mapping class groups act trivially if and only if for the canonical end the ribbon twist and double braiding with itself are trivial. We give a similar result for the visibility of the Torelli groups.