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Motoo Tange

Publications and source records attributed to Motoo Tange.

At least 19 recordsLinked to original sources
arXiv2025

Exotic elliptic surfaces without 1-handles

In this article, we consider a sufficient condition that a knot-surgery or log-transformation of $E(n)$ admits a handle decomposition without 1-handles. We show that if $K$ is a knot that the bridge number is $b(K)\le 9n$, then the knot-surgery $E(n)_K$ of the elliptic surface $E(n)$ admits a handle decomposition without 1-handles. This means that if $\gcd(p,q)=1$, and $\min\{p,q\}\le 9$, then $E(1)_{p,q}$ admits a handle decomposition without 1-handles. We also show that if $\gcd (p,q)=1$, $\min\{p,q\}\le 4$, then the double log-transformation $E(n)_{p,q}$ admits a handle decomposition without 1-handles for any positive integer $n$.

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arXiv2022

Pochette surgery of 4-sphere

Iwase and Matsumoto defined `pochette surgery' as a cut-and-paste on 4-manifolds along a 4-manifold homotopy equivalent to $S^2\vee S^1$. The first author in [10] studied infinitely many homotopy 4-spheres obtained by pochette surgery. In this paper we compute the homology of pochette surgery of any homology 4-sphere by using `linking number' of a pochette embedding. We prove that pochette surgery with the trivial cord does not change the diffeomorphism type or gives a Gluck surgery. We also show that there exist pochette surgeries on the 4-sphere with a non-trivial core sphere and a non-trivial cord such that the surgeries give the 4-sphere.

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arXiv2021

Upsilon invariants of L-space cable knots

We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,Υ_K$ and $Υ_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.

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arXiv2021

Non-existence theorems on infinite order corks

Suppose that $X,X'$ are simply-connected closed exotic 4-manifolds. It is well-known that $X'$ is obtained by an order 2 cork twist of $X$. We give an infinite exotic family of 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4-dimensional OS-invariants for a family to be generated by an infinite order cork and give non-contractible relatively exotic 4-manifolds that are never induced by any cork. Furthermore, we prove an estimate of the number of OS-invariants of 4-manifolds generated by a cork.

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arXiv2020

The third term in lens surgery polynomials

It is well-known that the second coefficient of the Alexander polynomial of any lens space knot in $S^3$ is $-1$. We show that the non-zero third coefficient condition of the Alexander polynomial of a lens space knot $K$ in $S^3$ confines the surgery to the one realized by the $(2,2g+1)$-torus knot, where $g$ is the genus of $K$. In particular, such a lens surgery polynomial coincides with $Δ_{T(2,2g+1)}(t)$.

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arXiv2019

Notes on Gompf's infinite order corks

For any positive integer $n$ we give a ${\mathbb Z}^n$-cork with a ${\mathbb Z}^n$-effective embedding in a 4-manifold being homeomorphic to $E(n)$. This means that a cork gives a subset ${\mathbb Z}^n$ in the differential structures on $E(n)$. Further, we describe handle decompositions of the twisted doubles (homotopy $S^4$) of Gompf's infinite order corks and show that they are Gluck twists and log transforms of $S^4$.

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arXiv2018

Homology spheres with $E_8$-fillings and arbitrarily large correction terms

In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to $-E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sub-lattice of definite filling and the maximal rank of even definite filling is arbitrarily large.

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arXiv2018

On the Alexander polynomial of lens space knot

Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's property as the existence of simple curves included in a region in ${\Bbb R}^2$. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This curve is much useful to obtain constraints of Alexander polynomials of lens space knots. For example, we can investigate the location of the second, third and fourth non-zero coefficients. The curve extracts new invariant $α$-index. The invariant is an important factor to determine Alexander polynomial of lens space knot. We classify lens space surgeries that the Alexander polynomial is the same as a $(2,r)$-torus knot and lens space surgeries with small genus and so on. As well as lens space knots in $S^3$, we also deal with lens space knots in homology spheres, which the surgery duals are simple (1,1)-knots.

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arXiv2018

Homology spheres yielding lens spaces

It is known by the author that there exist 20 families of Dehn surgeries in the Poincaré homology sphere yielding lens spaces. In this paper, we give the concrete knot diagrams of the families and extend them to families of lens space surgeries in Brieskorn homology spheres. We illustrate families of lens space surgeries in $Σ(2,3,6n\pm1)$ and $Σ(2,2s+1,2(2s+1)n\pm1)$ and so on. As other examples, we give lens space surgeries in graph homology spheres, which are obtained by splicing two Brieskorn homology spheres.

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arXiv2018

Ribbon disks with the same exterior

We construct an infinite family of slice disks with the same exterior, which gives an affirmative answer to an old question asked by Hitt and Sumners in 1981. Furthermore, we prove that these slice disks are ribbon disks.

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arXiv2017

Boundary-sum irreducible finite order corks

We prove for any positive integer $n$ there exist boundary-sum irreducible ${\mathbb Z}_n$-corks with Stein structure. Here `boundary-sum irreducible' means the manifold is indecomposable with respect to boundary-sum. We also verify that some of the finite order corks admit hyperbolic boundary by HIKMOT.

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arXiv2016

Finite order corks

We show that for any po sitive integer $m$, there exist order $n$ Stein corks. The boundaries are cyclic branched covers of slice knots embedded in the boundary of corks. By applying these corks to generalized forms, we give a method producing examples of many finite order corks, which are possibly not Stein cork.

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arXiv2015

Variations of 4-dimensional twists obtained by an infinite order plug

In the previous paper the author defined an infinite order plug $(P,φ)$ which gives rise to infinite Fintushel-Stern's knot-surgeries. Here, we give two 4-dimensional infinitely many exotic families $Y_n$, $Z_n$ of exotic enlargements of the plug. The families $Y_n$, $Z_n$ have $b_2=3$, $4$ and the boundaries are 3-manifolds with $b_1=1$, $0$ respectively. We give a plug (or g-cork) twist $(P,φ_{p,q})$ producing the 2-bridge knot or link surgery by combining the plug $(P,φ)$. As a further example, we describe a 4-dimensional twist $(M,μ)$ between knot-surgeries for two mutant knots. The twisted double concerning $(M,μ)$ gives a candidate of exotic $\#^2S^2\times S^2$.

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arXiv2015

Heegaard Floer homology of Matsumoto's manifolds

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$ if $n<2τ(K_2)$ holds. We also give a formula of Ozsváth-Szabó's $τ$-invariant as the total sum of the Euler numbers of the reduced filtration. We compute the $δ$-invariants of the twisted Whitehead doubles of torus knots and correction terms of the branched covers of the Whitehead doubles. By using Owens and Strle's obstruction we show that the $12$-twisted Whitehead double of the $(2,7)$-torus knot and the $20$-twisted Whitehead double of the $(3,7)$-torus knot are not slice but the double branched covers bound rational homology 4-balls. These are the first examples having a gap between sliceness and rational 4-ball bound-ness of the double branched cover.

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arXiv2015

A construction of slice knots via annulus twists

We give a new construction of slice knots via annulus twists. The simplest slice knots obtained by our method are those constructed by Omae. In this paper, we introduce a sufficient condition for given slice knots to be ribbon, and prove that all Omae's knots are ribbon.

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arXiv2014

The $E_8$-boundings of homology spheres and negative sphere classes in $E(1)$

We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are the maximal and minimal second Betti number divided by $8$ among definite spin boundings of a homology sphere. The similar invariants $g_8$ and $\overline{g_8}$ are defined by the maximal (or minimal) product sum of $E_8$-form of bounding 4-manifolds. We compute these invariants for some homology spheres. We construct $E_8$-boundings for some of Brieskorn 3-spheres $Σ(2,3,12n+5)$ by handle decomposition. As a by-product of the construction, some negative classes which consist of addition of several fiber classes plus one sectional class in $E(1)$ are represented by spheres.

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arXiv2014

Non-orientable genus of a knot in punctured $\mathbb{C}P ^2$

For any knot $K$ which bounds non-orientable and null-homologous surfaces $F$ in punctured $n\mathbb{C}P^2$, we construct a lower bound of the first Betti number of $F$ which consists of the signature of $K$ and the Heegaard Floer $d$-invariant of the integer homology sphere obtained by $1$-surgery along $K$. By using this lower bound, we prove that for any integer $k$, a certain knot cannot bound any surface which satisfies the above conditions and whose first Betti number is less than $k$.

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