吉田 純

We discuss twists on Frobenius algebras in the context of link homology. In his paper in 2006, Khovanov asserted that a twist of a Frobenius algebra yields an isomorphic chain complex on each link diagram. Although the result has been widely accepted for nearly two decades, a subtle gap in the original proof was found in the induction step of the construction of the isomorphism. Following discussion with Khovanov, we decided to provide a new proof. Our proof is based on a detailed analysis of configurations of circles in each state.

The goal of this paper is to prove a categorified analogue of Kontsevich's $4T$ relation on Vassiliev derivatives of Khovanov homology.

Khovanov homology extends to singular links via a categorified analogue of Vassiliev skein relation. In view of Vassiliev theory, the extended Khovanov homology can be seen as Vassiliev derivatives of Khovanov homology. In this paper, we develop a new method to compute the first derivative. Namely, we introduce a complex, called a crux complex, and prove that the Khovanov homologies of singular links with unique double points are homotopic to cofibers of endomorphisms on crux complexes. Since crux complexes are actually small for some links, the result enables a direct computation of the first derivative of Khovanov homology. Furthermore, it together with a categorified Vassiliev skein relation provides a brand-new method for the computation of Khovanov homology. In fact, we apply the result to determine the Khovanov complexes of all twist knots in a universal way.

In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams of crossing change, which we call the "genus-one morphism." It is proved that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain an invariant for singular tangles that extending sl(2) tangle homology; examples include Lee homology, Bar-Natan homology, and Naot's universal Khovanov homology as well as Khovanov homology with arbitrary coefficients. We also verify that the invariant satisfies categorified analogues of Vassiliev skein relation and the FI relation.

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the "genus-one" operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

We propose a new model for multicategories with symmetries with respect to Zhang's group operads. The fully faithful embedding of the category of group operads into that of crossed interval groups is made use of, and it is shown that every multicategory gives rise to a fibration, in a sense, over a quotient of the total category of group operads. The symmetric structures can be presented as structures of internal presheaves over a category internal to the category of small categories, in other words a double category.

The goal of the paper is to establish and to investigate a fully faithful embedding of the category of group operads into that of crossed interval groups. For this, we introduce a monoidal structure on the slice of the category of operads over the operad of symmetric groups. Comparing with the monoidal structure on the category of interval sets discussed in the author's previous work, we obtain a monoidal functor connecting these two categories. It will be shown that this actually induces a fully faithful functor on monoid objects and does not change the underlying sets, so we obtain a required embedding. The conditions for crossed interval groups to belong to the essential image will be proposed; namely in terms of commutativity of certain elements. As a result, it will turn out that the group operads form a reflective subcategory of the category of crossed interval groups. Finally, we will discuss monoid objects in symmetric monoidal category and Hochschild homologies on them.

Although the notion of crossed groups was originally introduced only in the simplicial case, the definition makes sense in the other categories. For instance, Batanin and Markl studied crossed interval groups to investigate symmetries on the Hochschild cohomology in view of operads. The aim of this paper is to make a comprehensive understanding of crossed groups for arbitrary base categories. In particular, we focus on the local presentability of the category of crossed groups, monadicity, and the basechange theorem along certain sorts of functors. The paper also contains the classification of crossed interval groups, which Batanin and Markl concerned about.

In this paper, we aim to provide a notion of "relative objects", i.e. objects equipped with some sort of subobjects, in differential topology. In spite of active researches relating them, e.g. knot theory or the theory of manifolds with corners, there seem to be poor general notions to deal with them. Moreover, we want even more direct differential calculus on relative objects and extension of classical notions and theories to relative situations; e.g. functions, vector fields, jet bundles, singularities, and so on. To establish this, the notion of arrangements of manifolds is introduced, which, for example, enables us to control behaviors of smooth maps on manifolds around corners. We construct jet bundles and prove a relative version of Transversality Theorem for some sorts of arrangements. Finally, embedding theorem of manifolds with faces into polyhedra is proved as an application.

In this paper, we introduce a method to construct new categories which look like "cubes", and discuss model structures on the presheaf categories over them. First, we introduce a notion of thin-powered structure on small categories, which provides a generalized notion of "power-sets" on categories. Next, we see that if a small category $\mathcal{R}$ admits a good thin-powered structure, we can construct a new category $\square(\mathcal{R})$ called the cubicalization of the category. We also see that $\square(\mathcal{R})$ is equipped with enough structures so that many arguments made for the classical cube category $\square$ are also available. In particular, it is a test category in the sense of Grothendieck. The resulting categories contain the cube category $\square$, the cube category with connections $\square^c$, the extended cubical category $\square_\Sigma$ introduced by Isaacson, and cube categories $\square_G$ symmetrized by more general group operads $G$. We finally discuss model structures on the presheaf categories $\square(\mathcal{R})^\wedge$ over cubicalizations. We prove that $\square(\mathcal{R})^\wedge$ admits a model structure such that the simplicial realization $\square(\mathcal{R})^\wedge\to SSet$ is a left Quillen functor. Moreover, in the case of $\square_G$ for group operads $G$, $\square^\wedge_G$ is a monoidal model category, and we have a sequence of monoidal Quillen equivalences $\square Set \to \square_G^\wedge\to SSet$. For example, if $G=B$ is the group operad consisting of braid groups, the category $\square^\wedge_B$ is a braided monoidal model category whose homotopy category is equivalent to that of $SSet$.