arXiv subjects
Publications and source records attributed to Masaaki Suzuki.
We explicitly determine the first Galois obstruction in the cokernel of the Johnson homomorphism of the mapping class group of a surface of genus $g$, for every genus $g \ge 2$. It is described as a sum of two terms which are considerably different in character. One lies in the kernel of the Enomoto-Satoh trace map, whereas the other belongs to a certain ideal which vanishes upon passage to the closed surface case.
In this paper, we discuss twisted Alexander polynomials of a knot for group extensions of a finite group in two directions. Firstly, we provide a mod $p$ formula for the twisted Alexander polynomial of a knot in the $3$-sphere associated with the regular representation of a finite group. Secondly, we consider twisted Alexander polynomials of a knot for a series of central extensions of a finite group. Moreover, we apply these formulas for twisted Alexander polynomials to the study of twisted Alexander vanishing groups and orders for non-fibered knots.
In our previous work, we introduced the notion of a twisted Alexander vanishing (TAV) group, defined as a finite group for which the corresponding twisted Alexander polynomial of a knot vanishes. In this paper, we discuss the orders of TAV groups and construct knots whose twisted Alexander polynomials vanish. Moreover, we show that every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.
In our previous work, we introduced the notion of the twisted Alexander vanishing order of knots, defined as the order of the smallest finite group for which the corresponding twisted Alexander polynomial vanishes. In this paper, we explore several properties of this invariant in detail and present a list of twisted Alexander vanishing groups of order less than $201$.
Clasper surgery induces the $Y$-filtration $\{Y_n\mathcal{IC}\}_n$ over the monoid of homology cylinders, which serves as a $3$-dimensional analogue of the lower central series of the Torelli group of a surface. In this paper, we investigate the torsion submodules of the associated graded modules of these filtrations. To detect torsion elements, we introduce a homomorphism on $Y_n\mathcal{IC}/Y_{n+1}$ induced by the degree $n+2$ part of the LMO functor. Additionally, we provide a formula that computes this homomorphism under clasper surgery, and use it to demonstrate that every non-trivial torsion element in $Y_6\mathcal{IC}/Y_7$ has order $3$.
Based on a vanishing theorem for non-fibered knots due to Friedl and Vidussi, we define the twisted Alexander vanishing order of a knot to be the order of the smallest finite group such that the corresponding twisted Alexander polynomial is zero. In this paper, we show its basic properties, and provide several explicit values for knots with $10$ or fewer crossings. Moreover, we characterize a finite group admitting the zero-twisted Alexander polynomial.
It is well-known that a knot is Fox $n$-colorable for a prime $n$ if and only if the knot group admits a surjective homomorphism to the dihedral group of degree $n$. However, this is not the case for links with two or more components. In this paper, we introduce a two-tone coloring on a link diagram, and give a condition for links so that the link groups admit surjective representations to the dihedral groups. In particular, it is shown that the link group of any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree.
The twisted Alexander polynomial of a knot is defined associated to a linear representation of the knot group. If there exists a surjective homomorphism of a knot group onto a finite group, then we obtain a representation of the knot group by the composition of the surjective homomorphism and the regular representation of the finite group. In this paper, we provide several formulas of the twisted Alexander polynomial of a knot associated to such representations in terms of the Alexander polynomial.
In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average braid index of all two-bridge knots with a given crossing number.
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 $\mathbb{Q}π_1Σ_{g,1}$, and prove that its reduction to $\widehat{\mathbb{Q}π_1Σ_{g,1}}/\hat{I}^{d+1}$ 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.
A lower bound of the Gordian distance is presented in terms of the Blanchfield pairing. Our approach, in particular, allows us to show at least for 195 pairs of unoriented nontrivial prime knots with up to 10 crossings that their Gordian distance is equal to 3, most of which are difficult to treat otherwise.
In this paper, we determine the average genus of all the $2$-bridge knots with a given crossing number. As a consequence, we obtain the oblique asymptote of this value as the crossing number grows.
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 $n \geq 2$.
We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking account into genus as well as crossing number. Next, we determine the number of epimorphisms between fibered $2$-bridge knot groups. Moreover, we discuss degree one maps and $2$-bridge knots uknotting number one.
We construct a series of homomorphisms from the $Y$-filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient $Y_3\mathcal{IC}_{g,1}/Y_4$ of the $Y$-filtration.
In the late 1980's, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration, and that its core part was identified with the secondary characteristic class $d_1$ associated with the fact that the first $\mathrm{MMM}$ class vanishes on the Torelli group (however it turned out that Johnson proved the former part highly likely prior to the above, see Remark 1.1). This secondary class $d_1$ is a rational generator of $H^1(\mathcal{K}_g;\mathbb{Z})^{\mathcal{M}_g}\cong\mathbb{Z}$ where $\mathcal{K}_g$ denotes the Johnson subgroup of the mapping class group $\mathcal{M}_g$. Hain proved, as a particular case of his fundamental result, that this is the only difference in degree $2$. In this paper, we prove that no other invariant than the above gives rise to new rational difference between the two filtrations up to degree $6$. We apply this to determine $H_1(\mathcal{K}_g;\mathbb{Q})$ explicitly by computing the description given by Dimca, Hain and Papadima. We also show that any finite type rational invariant of homology $3$-spheres of degrees up to $6$, including the second and the third Ohtsuki invariants, can be expressed by $d_1$ and lifts of Johnson homomorphisms.
Let $K,K'$ be two-bridge knots of genus $n,k$ respectively. We show the necessary and sufficient condition of $n$ in terms of $k$ that there exists an epimorphism from the knot group of $K$ onto that of $K'$.
We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces.