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Masaharu Ishikawa

Publications and source records attributed to Masaharu Ishikawa.

At least 19 recordsLinked to original sources
arXiv2026

Homogeneous Milnor fibers and Kato--Matsumoto bounds via simplicial multiwedges

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato--Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

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arXiv2026

Topology of the links of cDV singularities of types $cA_n$ for $n>0$ and $cD_n$ for $n>4$

We show that the second integral homology group of the link of an isolated compound Du Val (cDV, for short) singularity of type $cA_n$ is either trivial or a torsion-free abelian group. Consequently, by a result of Smale, it follows that the link is either $S^5$ or a connected sum of finitely many copies of $S^2\times S^3$. We also determine the rank of the second integral homology group of the link of a singularity of type $cD_n$ with $n>4$ under the assumption that the singularity is Newton non-degenerate. Furthermore, we focus on the weighted homogeneous case and determine the homology group of the link, including its torsion subgroup, under the assumption that the singularity is a Thom-Sebastiani sum of singularities of Brieskorn-Pham, cyclic, or chain type.

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arXiv2026

New $μ$-Zariski pairs of surface singularities

To the best of the authors' knowledge, all previously known examples of $μ$-, $μ^*$-, link-, or ordinary Zariski pairs of surface singularities in $\mathbb{C}^3$ consist of (possibly weighted) Lê-Yomdin singularities. In this paper, we present an example of a $μ$-Zariski pair involving surface singularities that are not of Lê-Yomdin type.

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arXiv2024

Twist spun knots of twist spun knots of classical knots

A $k$-twist spun knot is an $n+1$-dimensional knot in the $n+3$-dimensional sphere which is obtained from an $n$-dimensional knot in the $n+2$-dimensional sphere by applying an operation called a $k$-twist-spinning. This construction was introduced by Zeeman in 1965. In this paper, we show that the $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot is a trivial $3$-knot in $S^5$ if $\gcd(m_1,m_2)=1$. We also give a sufficient condition for the $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot to be non-trivial.

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arXiv2024

Distinguishing 2-knots admitting circle actions by fundamental groups

A 2-sphere embedded in the 4-sphere invariant under a circle action is called a branched twist spin. A branched twist spin is constructed from a 1-knot in the 3-sphere and a pair of coprime integers uniquely. In this paper, we study, for each pair of coprime integers, if two different 1-knots yield the same branched twist spin, and prove that such a pair of 1-knots does not exist in most cases. Fundamental groups of 3-orbifolds of cyclic type are obtained as quotient groups of the fundamental groups of the complements of branched twist spins. We use these groups for distinguishing branched twist spins.

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arXiv2024

Atypical values at infinity of real polynomial maps with $2$-dimensional fibers

We characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is an analogy of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are $2$-dimensional surfaces.

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arXiv2024

Combinatorial cusp count and clover invariants

We construct efficient topological cobordisms between torus links and large connected sums of trefoil knots. As an application, we show that the signature invariant $σ_ω$ at $ω=ζ_6$ takes essentially minimal values on torus links among all concordance homomorphisms with the same normalisation on the trefoil knot.

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arXiv2023

Relative homotopy groups and Serre fibrations for polynomial maps

Let $f$ be a polynomial map from $\mathbb R^m$ to $\mathbb R^n$ with $m>n>0$ and $t_0$ be a regular value of $f$. For a small open ball $D_{t_0}$ centered at $t_0$, we show that the map $f:f^{-1}(D_{t_0})\to D_{t_0}$ is a Serre fibration if and only if $f$ is a Serre fibration over a finite number of certain simple arcs starting at $t_0$. We characterize the fibration $f:f^{-1}(D_{t_0})\to D_{t_0}$ by relative homotopy groups defined for these arcs and use it to prove the assertion.

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arXiv2023

Non-integral boundary slopes of alternating knots

We show, for every positive integer $n$, there is an alternating knot having a boundary slope with denominator $n$. We make use of Kabaya's method for boundary slopes and the layered solid torus construction introduced by Jaco and Rubinstein and further developed by Howie et al.

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arXiv2023

Positive flow-spines and contact 3-manifolds, II

This paper corresponds to Section 8 of arXiv:1912.05774v3 [math.GT]. The contents until Section 7 are published in Annali di Matematica Pura ed Applicata as a separate paper. In that paper, it is proved that for any positive flow-spine P of a closed, oriented 3-manifold M, there exists a unique contact structure supported by P up to isotopy. In particular, this defines a map from the set of isotopy classes of positive flow-spines of M to the set of isotopy classes of contact structures on M. In this paper, we show that this map is surjective. As a corollary, we show that any flow-spine can be deformed to a positive flow-spine by applying first and second regular moves successively.

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arXiv2023

Positive flow-spines and contact 3-manifolds

A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form whose Reeb flow is a flow of the flow-spine. It is known by Thurston and Winkelnkemper that any open book decomposition of a closed oriented 3-manifold supports a contact structure. In this paper, we introduce a notion of positivity for flow-spines and prove that any positive flow-spine of a closed, connected, oriented 3-manifold supports a contact structure uniquely up to isotopy. The positivity condition is critical to the existence of the unique, supported contact structure, which is also proved in the paper.

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arXiv2022

Presentation of the fundamental groups of complements of shadows

A shadowed polyhedron is a simple polyhedron equipped with half integers on regions, called gleams, which represents a compact, oriented, smooth 4-manifold. The polyhedron is embedded in the 4-manifold and it is called a shadow of that manifold. A subpolyhedron of a shadow represents a possibly singular subsurface in the 4-manifold. In this paper, we focus on contractible shadows obtained from the unit disk by attaching annuli along generically immersed closed curves on the disk. In this case, the 4-manifold is always a 4-ball. Milnor fibers of plane curve singularities and complexified real line arrangements can be represented in this way. We give a presentation of the fundamental group of the complement of a subpolyhedron of such a shadow in the 4-ball. The method is very similar to the Wirtinger presentation of links in knot theory.

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arXiv2020

S-stable foliations on flow-spines with transverse Reeb flow

The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $β$ defining a foliation on a branched simple polyhedron $P$ satisfies $dβ>0$, which means that the foliation is a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $β$ on $P$ with $dβ>0$ then we can find a 1-form with the same property and additionally being S-stable. We then prove that the number of simple tangency points of an S-stable foliation on a positive or negative flow-spine is at least 2 and give a recipe for constructing a characteristic foliation of a 1-form $β$ with $dβ>0$ on the abalone.

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arXiv2020

Milnor fibration, A'Campo's divide and Turaev's shadow

We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which we call the LF-property. We will show that the shadowed polyhedron constructed from a divide satisfies this property and the Lefschetz fibration of this polyhedron is isomorphic to the Lefschetz fibration of the divide. Furthermore, applying the same technique to certain free divides we will show that the links of those free divides are fibered with positive monodromy.

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arXiv2017

Virtual unknotting numbers of certain virtual torus knots

The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the $(p,q)$-torus knot, i.e. it is (p-1)(q-1)/2.

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arXiv2016

The bifurcation set of a real polynomial function of two variables and Newton polygons of singularities at infinity

In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called "cleaving" and "vanishing" in the same setting. Finally, we give an upper bound of the number of elements in the bifurcation set in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the singularities at infinity successively.

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