arXiv subjects

Wataru Yuasa

Publications and source records attributed to Wataru Yuasa.

13 recordsLinked to original sources

arXiv2025

Bounded $\mathfrak{sp}_4$-laminations and their intersection coordinates

We introduce rational bounded $\mathfrak{sp}_4$-laminations on a marked surface $\boldsymbol{\Sigma}$ as a proposed topological model for the rational tropical points $\mathcal{A}_{Sp_4,\boldsymbol{\Sigma}}(\mathbb{Q}^{\mathsf{T}})$ of the Fock--Goncharov moduli space [FG06]. Our space consists of certain equivalence classes of $\mathfrak{sp}_4$-webs introduced by Kuperberg [Kup96], together with rational measures. We define tropical coordinate systems using the $\mathfrak{sp}_4$-case of the intersection number of Shen--Sun--Weng [SSW25], and establish a bijection using the framework of the graded $\mathfrak{sp}_4$-skein algebra. This provides a topological perspective for Fock--Goncharov duality for $\mathfrak{sp}_4$.

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arXiv2023

Skein and cluster algebras with coefficients for unpunctured surfaces

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum cluster structure. The walled surfaces naturally generalize the marked surfaces with multi-laminations, which have been used to describe the quantum cluster algebras of geometric type for marked surfaces by Fomin--Thurston [FT18]. Moreover, we give skein theoretic interpretation for some of quasi-homomorphisms [Fra16] between these quantum cluster algebras.

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arXiv2022

Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$

Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$-case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $\Sigma$ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural $\mathbb{Z}_q$-form $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{\mathbb{Z}_q} \subset \mathscr{S}_{\mathfrak{sp}_4,\Sigma}^q$, while the natural coefficient ring $\mathcal{R}$ of $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^q$ includes the inverse of the quantum integer $[2]_q$. We prove that its boundary-localization $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{\mathbb{Z}_q}[\partial^{-1}]$ is included into a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sp}_4,\Sigma}$ that quantizes the function ring of the moduli space $\mathcal{A}_{Sp_4,\Sigma}^\times$. Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to [IY21](arXiv:2101.00643). We also propose a characterization of cluster variables in the spirit of Fomin--Pylyavksyy [FP16](arXiv:1210.1888) in terms of the $\mathfrak{sp}_4$-webs, and give infinitely many supporting examples on a quadrilateral.

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arXiv2021

Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sl}_3$

For an unpunctured marked surface $\Sigma$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sl}_3,\Sigma}$ inside the skew-field $\mathrm{Frac}\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ of fractions, which quantizes the cluster $K_2$-structure on the moduli space $\mathcal{A}_{SL_3,\Sigma}$ of decorated $SL_3$-local systems on $\Sigma$. We show that the cluster algebra $\mathscr{A}^q_{\mathfrak{sl}_3,\Sigma}$ contains the boundary-localized skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}[\partial^{-1}]$ as a subalgebra, and their natural structures, such as gradings and certain group actions, agree with each other. We also give an algorithm to compute the Laurent expressions of a given $\mathfrak{sl}_3$-web in certain clusters and discuss the positivity of coefficients. In particular, we show that the bracelets and the bangles along an oriented simple loop in $\Sigma$ have Laurent expressions with positive coefficients, hence give rise to quantum GS-universally positive Laurent polynomials.

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arXiv2020

The zero stability for the one-row colored $\mathfrak{sl}_3$ Jones polynomial

The stability of coefficients of colored ($\mathfrak{sl}_2$-) Jones polynomials $\{J_{K,n}^{\mathfrak{sl}_2}(q)\}_n$ was discovered by Dasbach and Lin. This stability is now called the zero-stability of $J_{K,n}^{\mathfrak{sl}_2}(q)$. Armond showed zero stability for a $B$-adequate link by using the linear skein theory based on the Kauffman bracket. In this paper, we prove the zero stability of one-row colored $\mathfrak{sl}_{3}$-Jones polynomials $\{J_{K,n}^{\mathfrak{sl}_3}(q)\}_n$ for $B$-adequate links $L$ with anti-parallel twist regions by using the linear skein theory based on Kuperberg's $\mathfrak{sl}_3$-webs. It implies the existence of many $q$-series obtained from a quantum invariant associated with $\mathfrak{sl}_3$.

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arXiv2020

Twist formulas for one-row colored $A_2$ webs and $\mathfrak{sl}_3$ tails of $(2,2m)$-torus links

The $\mathfrak{sl}_3$ colored Jones polynomial $J_{\lambda}^{\mathfrak{sl}_3}(L)$ is obtained by coloring the link components with two-row Young diagram $\lambda$. Although it is difficult to compute $J_{\lambda}^{\mathfrak{sl}_3}(L)$ in general, we can calculate it by using Kuperberg's $A_2$ skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in $A_2$ web space and compute $J_{(n,0)}^{\mathfrak{sl}_3}(T(2,2m))$ for an oriented $(2,2m)$-torus link. These explicit formulas derives the $\mathfrak{sl}_3$ tail of $T(2,2m)$. They also give explicit descriptions of the $\mathfrak{sl}_3$ false theta series with one-row coloring because the $\mathfrak{sl}_2$ tail of $T(2,2m)$ is known as the false theta series.

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arXiv2019

A graphical categorification of the two-variable Chebyshev polynomials of the second kind

We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q(\mathfrak{sl}_3)$ and the $A_2$ clasps are projectors. Our categorification also gives a natural definition of a $q$-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.

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arXiv2018

On power subgroups of Dehn twists in hyperelliptic mapping class groups

This paper contains two topics, the index of a power subgroup in the mapping class group $\mathcal{M}(0,2n)$ of a $2n$-punctured sphere and in the hyperelliptic mapping class group $\Delta(g,0)$ of an oriented closed surface of genus $g$. The main tool is a projective representation of $\mathcal{M}(0,2n)$ obtained through the Kauffman bracket skein module. For $\mathcal{M}(0,2n)$, we prove that the normal closure of the fifth power of a half-twist has infinite index. This is the remaining case of a Masbaum's work. For $\Delta(g,0)$, we consider the normal closure of $m$-th powers of Dehn twists along all symmetric simple closed curves. We show the subgroup has infinite index if $m\geq 5$ and $m\neq 6$ for any $g\geq 2$.

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arXiv2017

$A_2$ Skein Representations of Pure Braid Groups

We define a family of representations $\{\rho_n\}_{n\geq 0}$ of a pure braid group $P_{2k}$. These representations are obtained from an action of $P_{2k}$ on a certain type of $A_2$ web space with color $n$. The $A_2$ web space is a generalization of the Kauffman bracket skein module of a disk with marked points on its boundary. We also introduce a triangle-free basis of such an $A_2$ web space and calculate matrix representations of $\rho_n$ about the standard generators of $P_{2k}$.

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arXiv2017

$A_2$ colored polynomials of rigid vertex graphs

The Kauffman-Vogel polynomials are three variable polynomial invariants of $4$-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented $4$-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with $2$. Bataineh, Elhamdadi and Hajij generalized it to any color with even positive integers. We give another generalization of the one-variable Kauffman-Vogel polynomial for oriented and unoriented $4$-valent rigid vertex graphs by using the $A_2$ bracket and the $A_2$ clasps. These polynomial invariants are considered as the $\mathfrak{sl}_3$ colored Jones polynomials for singular knots and links.

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arXiv2016

A $q$-series identity via the $\mathfrak{sl}_3$ colored Jones polynomials for the $(2,2m)$-torus link

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathfrak{sl}_3$ colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the $\mathfrak{sl}_3$ colored Jones polynomials colored by $(n,0)$ for the $(2,2m)$-torus link. These two expressions of the tail provide an identity of $q$-series. This is a knot-theoretical generalization of the Andrews-Gordon identities for the Ramanujan false theta function.

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arXiv2016

The $\mathfrak{sl}_3$ colored Jones polynomials for $2$-bridge links

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ colored Jones polynomials of $2$-bridge knots and links explicitly using twist formulas.

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arXiv2015

Poisson algebras of curves on bordered surfaces and skein quantization

We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quotient of the vector space spanned by the regular homotopy classes of curves on the bordered surface by generalizing the Goldman bracket and the Turaev cobracket. Moreover, we define a Poisson algebra of unoriented curves on a bordered surface and show that a quantization of the Poisson algebra coincides with the skein algebra of the bordered surface defined by Muller.

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