arXiv subjects

Kazunori Iwaki

Publications and source records attributed to Kazunori Iwaki.

4 recordsLinked to original sources

arXiv2026

Cappell-Shaneson knot pairs with the same Alexander polynomial

It is well known that for $m\geq 2$ there are at most two non-equivalent $m$-knots with diffeomorphic exterior. Such pair of knots will be called $\textit{ non-reflexive knot pair}$. A classical problem in topology is to determine all dimensions where such knot pairs exist. In 1976 Cappell and Shaneson gave a method of constructing non-reflexive knot pairs. In the present paper we construct an infinite family of new examples of Cappell-Shaneson knot pairs, and give examples of Cappell-Shaneson knot pairs that have the same Alexander polynomial but are inequivalent.

math.GT↗

arXiv2025

Cappell-Shaneson polynomials

S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial is the characteristic polynomial of a Cappell-Shaneson matrix. In this paper, we interpret part of the definition of Cappell-Shaneson polynomial as algebraic conditions on polynomials in terms of signed reciprocal polynomial and reduction modulo primes, and give complete lists of all Cappell-Shaneson polynomials of degrees $4$ and $5$. We construct several infinite series of Cappell-Shaneson polynomials of degree $6$.

math.GT↗

arXiv2024

Infinite families of standard Cappell-Shaneson spheres

Cappell-Shaneson homotopy 4-spheres (CS spheres) are potential counterexamples of the smooth 4-dimensional Poincaré conjecture. Akbulut proved that infinite CS spheres are diffeomorphic to the standard 4-sphere by Kirby calculus. Kim and Yamada found another family of CS spheres which is composed of standard CS spheres. In this paper, we prove more CS spheres are standard. We give 145 new infinite families of CS spheres which are diffeomorphic to the standard 4-sphere.

math.GT↗

arXiv2021

Neural Network Libraries: A Deep Learning Framework Designed from Engineers' Perspectives

While there exist a plethora of deep learning tools and frameworks, the fast-growing complexity of the field brings new demands and challenges, such as more flexible network design, speedy computation on distributed setting, and compatibility between different tools. In this paper, we introduce Neural Network Libraries (https://nnabla.org), a deep learning framework designed from engineer's perspective, with emphasis on usability and compatibility as its core design principles. We elaborate on each of our design principles and its merits, and validate our attempts via experiments.

cs.LG↗