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Takeyoshi Kogiso

Publications and source records attributed to Takeyoshi Kogiso.

8 recordsLinked to original sources
arXiv2026

Degree shifts between q-deformed friezes and q-Farey labelings for general triangulations

Morier-Genoud and Ovsienko introduced q-deformations of continued fractions, Farey labelings, and Conway--Coxeter friezes, and established relationships among them in restricted settings associated with triangulations having exactly two exterior cells. In this paper, we extend these correspondences to arbitrary subsequences of quiddities arising from general triangulations. We show that the numerator and denominator polynomials of q-deformed continued fractions coincide with entries of q-deformed Conway--Coxeter friezes, while the corresponding polynomials in q-Farey labelings agree with them up to explicit powers of q. These powers are described combinatorially in terms of the number of diagonals in the triangulation, or equivalently, the number of entries equal to 1 in the associated frieze. Furthermore, we determine the minimum and maximum degrees of these polynomials in terms of the same combinatorial data.

math.CO↗
arXiv2024

Arithmetic on $q$-deformed rational numbers

Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal{R}_{\frac{r}s}(q),~ \mathcal{S}_{\frac{r}s}(q) \in {\mathbb Z}[q]$ with $\mathcal{R}_{\frac{r}s}(1)=r,~\mathcal{S}_{\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{s}$, then $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal{S}_{\frac{r}s}(ζ)=0$ for $ζ=(-1+\sqrt{-3})/2$ (resp. $ζ=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links.

math.CO↗
arXiv2022

Prehomogeneous vector spaces obtained from triangle arrangements

In this paper, we construct a new series of prehomogeneous vector spaces from figures made up of triangles, called triangle arrangements. Our main theorem states that, under suitable assumptions, we are able to construct a prehomogeneous vector space obtained from a triangle arrangement by attaching two triangle arrangements corresponding to prehomogeneous vector spaces at a vertex. We also give examples of prehomogeneous vector spaces obtained from triangle arrangements. Many of them seem to be new.

math.RT↗
arXiv2020

$q$-Deformations and $t$-deformations of Markov triples

In this paper, we generalize the Markov triples in two different directions. One is generalization in direction of using the $q$-deformation of rational number introduced by \cite{MO} in connection with cluster algebras, quantum topology and analytic number theory. The other is direction using castling transforms of prehomogeneous vector spaces \cite{SaKi} which plays an important role in the study of representation theory and automorphic function. In addition, the present paper gives a relationship between the two generalizations. This may provide some kind of bridging between different fields.

math.NT↗
arXiv2020

A characterization of Conway-Coxeter friezes of zigzag type by rational links

The present paper show that Conway-Coxeter friezes of zigzag type are characterized by (unoriented) rational links. As an application of this characterization Jones polynomial can be defined for Conway-Coxeter friezes of zigzag type. This gives a new method for computing the Jones polynomial for oriented rational links.

math.GT↗
arXiv2016

Some properties of associated spaces with sub-Hankel determinants

In this note, we show that the space associated with sub-Hankel determinant is a non-reductive, regular prehomogeneous vector space, and we give the multiplicative Legendre transforms of sub-Hankel determinants. Moreover we observe certain relations between $b$-functions of polarization of PVpolynomials and $b$-functions of sub-Hankel determinants, and give some formulas about sub-Hankel determinants whose components are orthogonal ponlynomials.

math.RT↗