Finiteness of the image of the Reidemeister torsion of a splice

The set $\mathit{RT}(M)$ of values of the $\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion of a 3-manifold $M$ can be both finite and infinite. We prove that $\mathit{RT}(M)$ is a finite set if $M$ is the splice of two certain knots in the 3-sphere. The proof is based on an observation on the character varieties and $A$-polynomials of knots.Comment: 16 pages, 1 figure, to appear in Ann. Math. Blaise Pasca

On the kernel of the surgery map (Intelligence of Low-dimensional Topology)

A Jacobi diagram gives a clasper in the trivial homology cylinder, and then one obtains another homology cylinder by surgery along the clasper. This procedure defines a homomorphism Sn: A[c, n] → YnLCg, ₁/Yn+1 between abelian groups. Sato, Suzuki, and the author [15, 16] constructed a homomorphism on YnLCg, ₁/Yn+1, and gave an application to the study of the surgery map Sn. The purpose of this article is to review the results in [16] and introduce related works on the surgery map

On the kernel of the surgery map restricted to the 1-loop part

Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map $\mathfrak{s} \colon \mathcal{A}_n^c \to Y_n\mathcal{IC}_{g,1}/Y_{n+1}$ is surjective for $n \geq 2$, and its kernel is closely related to the symmetry of Jacobi diagrams. We determine the kernel of $\mathfrak{s}$ restricted to the 1-loop part after taking a certain quotient of the target. Also, we introduce refined versions of the AS and STU relations among claspers and study the abelian group $Y_n\mathcal{IC}_{g,1}/Y_{n+2}$ for $n \geq 2$.Comment: 33 pages, 4 figure

A non-commutative Reidemeister-Turaev torsion of homology cylinders

We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the $K_1$-group of the $I$-adic completion of the group ring $\mathbb{Q}\pi_1\Sigma_{g,1}$, and prove that its reduction to $\widehat{\mathbb{Q}\pi_1\Sigma_{g,1}}/\hat{I}^{d+1}$ is a finite-type invariant of degree $d$. We also show that the $1$-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion.Comment: 48 pages, 7 figure

ホモロジーコボルディズムを用いた３次元多様体の不変量とレンズ空間内の結び目

学位の種別: 課程博士審査委員会委員 : （主査）東京大学准教授 逆井 卓也, 東京大学教授 坪井 俊, 東京大学教授 河野 俊丈, 東京大学教授 古田 幹雄, 東京大学准教授 河澄 響矢, 東京大学准教授 北山 貴裕University of Tokyo(東京大学