Journal of Topology最新文献

We study the action of the orthogonal group on the little $�n$-disks operads. As an application we provide small models (over the reals) for the framed little $�n$-disks operads. It follows in particular that the framed little $�n$-disks operads are formal (over the reals) for $�n$ even and coformal for all $�n$.

We show that closures of homogeneous braids are visually prime, addressing a question of Cromwell. The key technical tool for the proof is the following criterion concerning primeness of open books, which we consider to be of independent interest. For open books of 3-manifolds the property of having no fixed essential arcs is preserved under essential Murasugi sums with a strictly right-veering open book, if the plumbing region of the original open book veers to the left. We also provide examples of open books in $�S^3$ demonstrating that primeness is not necessarily preserved under essential Murasugi sum, in fact not even under stabilizations also known as Hopf plumbings. Furthermore, we find that trefoil plumbings need not preserve primeness. In contrast, we establish that figure-eight knot plumbings do preserve primeness.

An essential goal in the study of finite-type invariants of some objects (knots, manifolds) is the construction of a universal finite-type invariant, universal in the sense that it contains all finite-type invariants of the given objects. Such a universal finite-type invariant is known for knots in the 3-sphere — the Kontsevich integral — and for homology 3-spheres — the Le–Murakami–Ohtsuki invariant. For knots in homology 3-spheres, an invariant constructed by Garoufalidis and Kricker as a lift of the Kontsevich integral has been considered for the last two decades as the best candidate to be a universal finite-type invariant. Although this invariant is eventually universal in restriction to knots whose Alexander polynomial is trivial, we prove here that it is not powerful enough in general. For that, we provide a refinement of its construction which produces a strictly stronger invariant, and we prove that this new invariant is a universal finite-type invariant of knots in homology 3-spheres. This provides a full diagrammatic description of the graded space of finite-type invariants of knots in homology 3-spheres.

We provide a lower bound on the volume of the compactified mapping torus of a strongly irreducible end-periodic homeomorphism $��f�:�S�\to �S�$. This result, together with work of Field, Kim, Leininger, and Loving [J. Topol. 16 (2023), no. 1, 57–105], shows that the volume of $�{\overline{M}}_f$ is comparable to the translation length of $�f$ on a connected component of the pants graph $��P�\left(�S�\right)�$, extending work of Brock [Comm. Anal. Geom. 11 (2003), no. 5, 987–999] in the finite-type setting on volumes of mapping tori of pseudo-Anosov homeomorphisms.

Let $��n�,�m�⩾�2�$. Let be a Zariski dense convex cocompact subgroup and $��\Lambda �\subset �S^n�$ be its limit set. Let $��\rho �:�\Gamma �\to �{\text{SO}}^{\circ }��(�m�+�1�,�1�)��$ be a Zariski dense convex cocompact faithful representation and $��f�:�\Lambda �\to �S^m�$ the $�\rho$-boundary map. Let

We refine several results of Bhatt–Morrow–Scholze on $�\mathrm{THH}$ to real topological Hochschild homology ($�\mathrm{THR}$). In particular, we compute $�\mathrm{THR}$ 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.

We establish the geometry behind the quantum $��6�j�$-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 $��6�j�$-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 $��6�j�$-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 $��6�j�$-symbols could decay exponentially. This is a phenomenon that has never been seen before.