The Bott–Cattaneo–Rossi invariant $(Z_k)_{k \in \mathbb{N} \setminus \{0, 1\}}$ is an invariant of long knots $\mathbb{R}^{n} \hookrightarrow \mathbb{R}^{n+2}$ for odd $n$, which reads as a combination of integrals over configuration spaces. In this article, we compute such integrals and prove explicit formulas for (generalized) $Z_{k}$ in terms of Alexander polynomials, or in terms of linking numbers of some cycles of a hypersurface bounded by the knot. Our formulas, which hold for all null-homologous long knots in homology $\mathbb{R}^{n+2}$ at least when $n \equiv 1 \mod 4$, conversely express the Reidemeister torsion of the knot complement in terms of $(Z_{k})_{k \in \mathbb{N} \setminus \{0, 1\}}$. Our formula extends to the even-dimensional case, where $Z_{k}$ will be proved to be well-defined in an upcoming article.

The Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.

We solve the problem of counting Jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

The notion of Zariski pairs for projective curves in $\mathbb{P}^{2}$ is known since the pioneer paper of Zariski. In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski) pair of curves $C = \{f(x, y, z) = 0\}$ and $C' = \{g(x, y, z) = 0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. They have a same zeta function and a Milnor number. We give new examples of weak Zariski pairs which have same $\mu^{*}$ sequences and same zeta functions but two functions belong to different connected components of $\mu$-constant strata (Theorem 14). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology, which implies the Jordan forms of their monodromies are different (Theorem 24). We start from weak Zariski pairs of projective curves to construct new Zariski pairs of surfaces which have non-diffeomorphic link 3-folds. We also prove that the hypersurface pair constructed from a Zariski pair of irreducible plane curves with simple singularities give a diffeomorphic links (Theorem 25).

We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of $X$ is at least 3, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.

We study Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. There are exactly 9 types of such surfaces.

Let $f$ be a primitive form with respect to $\mathrm{SL}_{2}(\mathbb{Z})$. Then, we propose a conjecture on the congruence between the Klingen–Eisenstein lift of the Duke–Imamoglu–Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $\mathrm{Sp}_{2}(\mathbb{Z})$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.

The region select game, introduced by Ayaka Shimizu, Akio Kawauchi and Kengo Kishimoto, is a game that is played on knot diagrams whose crossings are endowed with two colors. The game is based on the region crossing change moves that induce an unknotting operation on knot diagrams. We generalize the region select game to be played on a knot diagram endowed with $k$-colors at its vertices for $2 \leq k \leq \infty$.

We give a recursive formula for the Alexander polynomials of pretzel knots with a pair of integer parameters with opposite signs. Using the formula, we characterize certain pretzel knots which are simple-ribbon.