Nobuo Iida, Masaki Taniguchi
We present a framework for studying transverse knots and symplectic surfaces utilizing the Seiberg-Witten monopole equation. Our primary approach involves investigating an equivariant Seiberg-Witten theory introduced by Baraglia-Hekmati on branched covers, incorporating invariant contact/symplectic structures. Within this framework, we introduce a novel slice-torus invariant denoted as $q_M(K)$. This invariant can be viewed as the Seiberg-Witten analog of Hendricks-Lipshitz-Sarker's $q_τ$ invariant, with a signature correction term. One property of the invariant $q_M(K)$ is an adjunction equality for properly embedded connected symplectic surfaces in the symplectic filling $D^4\# m \overline{\mathbb{C}P}^2$. The proof of this equality utilizes the equivariant version of the homotopical contact invariant introduced by the authors, leading to a transverse knot invariant. Another ingredient of the proof involves constructing invariant symplectic structures on branched covering spaces branched along properly embedded symplectic surfaces in symplectic fillings. As an application of the invariant $q_M(K)$, we determine the value of any slice-torus invariant within a permissible deviation of $2$ for squeezed knots concordant to certain Montesinos knots. Additionally, we provide an obstruction to realizing second homology classes of $D^4 \#m \overline{\mathbb{C}P}^2$ as connected embedded symplectic surfaces with transverse knot boundary or connected embedded Lagrangian surfaces with collarable Legendrian knot boundary. Moreover, we introduce a new obstruction to certain Montesinos knots being quasipositive, which is described only in terms of slice genera and their signatures.
Masaki Taniguchi
We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer cohomology due to R.Fintushel-R.Stern. We make use of the $\mathbb{Z}$-fold covering space of homology $S^3\times S^1$ and the instantons on it.
Hokuto Konno, Masaki Taniguchi
We show $ 10/8$-type inequalities for some end-periodic $4$-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed $4$-manifolds which do not admit positive scalar curvature metrics.
Nobuo Iida, Masaki Taniguchi
We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozváth-Szabó. Moreover, we prove a gluing formula relating our invariant with the first author's Bauer-Furuta type invariant, which refines Kronheimer-Mrowka's invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.
Hokuto Konno, Jin Miyazawa, Masaki Taniguchi
We develop a version of Seiberg--Witten Floer cohomology/homotopy type for a spin$^c$ 4-manifold with boundary and with an involution which reverses the spin$^c$ structure, as well as a version of Floer cohomology/homotopy type for oriented links with non-zero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3-manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov-type inequalities which relate topological quantities of 4-manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, Nielsen realization problem for non-spin 4-manifolds, and non-smoothable unoriented surfaces in 4-manifolds.
Nobuo Iida, Hokuto Konno, Masaki Taniguchi
We give a generalized Thurston--Bennequin-type inequality for links in $S^3$ using a Bauer--Furuta-type invariant for 4-manifolds with contact boundary. As a special case, we also give an adjunction inequality for smoothly embedded orientable surfaces with negative intersection in a closed oriented smooth 4-manifold whose non-equivariant Bauer--Furuta invariant is non-zero.
Yuta Nozaki, Kouki Sato, Masaki Taniguchi
For any $s \in [-\infty, 0] $ and oriented homology 3-sphere $Y$, we introduce a homology cobordism invariant $r_s(Y)\in (0,\infty]$. The values $\{r_s(Y)\}$ are included in the critical values of the $SU(2)$-Chern-Simons functional of $Y$, and we show a negative definite cobordism inequality and a connected sum formula for $r_s$. As applications, we obtain several new results on the homology cobordism group. First, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. Next, we show that if the 1-surgery of $S^3$ along a knot has the Frøyshov invariant negative, then all positive $1/n$-surgeries along the knot are linearly independent in the homology cobordism group. In another direction, we use $\{r_s\}$ to define a filtration on the homology cobordism group which is parametrized by $[0,\infty]$. Moreover, we compute an approximate value of $r_s$ for the hyperbolic 3-manifold obtained by $1/2$-surgery along the mirror of the knot $5_2$.
Yoshihiro Fukumoto, Masaki Taniguchi
We show that the 3-fold (resp. 6-fold) connected sum of the $(2,1)$-cable of the figure-eight knot cannot bound a smooth null-homologous disk in a punctured $S^2 \times S^2$ (resp. in a punctured $#_2 S^2 \times S^2$. This result is obtained using a real version of the $10/8$-inequality established by Konno, Miyazawa, and Taniguchi.
Hokuto Konno, Anubhav Mukherjee, Masaki Taniguchi
In a small simply-connected closed 4-manifold, we construct infinitely many pairs of exotic codimension-$1$ submanifolds with diffeomorphic complements that remain exotic after any number of stabilizations by $ S^2 \times S^2$. We also give new constructions of exotic embeddings of 3-spheres in 4-manifolds with diffeomorphic complements.
Nobuo Iida, Anubhav Mukherjee, Masaki Taniguchi
Our main result gives an adjunction inequality for embedded surfaces in certain $4$-manifolds with contact boundary under a non-vanishing assumption on the Bauer--Furuta type invariants. Using this, we give infinitely many knots in $S^3$ that are not smoothly H-slice (that is, bounding a null-homologous disk) in many $4$-manifolds but they are topologically H-slice. In particular, we give such knots in the boundaries of the punctured elliptic surfaces $E(2n)$. In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with $b_3=0$ into closed symplectic 4-manifolds with $b_1=0$ and $b_2^+\equiv 3$ mod $4$. From here we prove a Bennequin type inequality for symplectic caps of $(S^3,ξ_{std})$. We also show that any weakly symplectically fillable $3$-manifold bounds a $4$-manifold with at least two smooth structures.
Masaki Taniguchi
For a given smooth $2$-knot in $S^4$, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible $ SU(2)$-representations of its knot group. For example, we see that any smooth $2$-knot having the Poincaré homology $3$-sphere as a Seifert hypersurface has at least four irreducible $SU(2)$-representations of its knot group. This result is false in the topological category. The proof uses a quantitative formulation of instanton Floer homology. Using similar techniques, we also obtain similar results about codimension-$1$ embeddings of homology $3$-spheres into closed definite $4$-manifolds and a fixed point type theorem for instanton Floer homology.
Nobuo Iida, Taketo Sano, Kouki Sato, Masaki Taniguchi
We show that Iida--Taniguchi's $\mathbb{Z}$-valued slice-torus invariant $q_M$ cannot be realized as a linear combination of Rasmussen's $s$-invariant, Ozsváth--Szabó's $τ$-invariant, all of the $\mathfrak{sl}_N$-concordance invariants ($N \geq 2$), Baldwin--Sivek's instanton $τ$-invariant, Daemi--Imori--Sato--Scaduto--Taniguchi's instanton $\tilde{s}$-invariant and Sano--Sato's Rasmussen type invariants $\tilde{ss}_c$.
Hayato Imori, Taketo Sano, Kouki Sato, Masaki Taniguchi
This paper is a continuation of our previous work, where we defined an embedded cobordism map on the instanton cube complex that recovers the cobordism maps both in Khovanov homology and singular instanton theory. In this paper, we extend this construction to immersed cobordisms, where we define an immersed cobordism map on Khovanov homology and prove that it is compatible with the immersed cobordism map on singular instanton homology. We give two applications: (i) For any smooth, oriented concordance $C$ from a two-bridge torus knot, the induced map $\mathit{Kh}(C)$ on Khovanov homology is injective, and its left inverse is given by the reversal of $C$. (ii) Any pair of relatively exotic surfaces in $D^4$ that are detected by the embedded cobordism map in $\mathit{Kh}$ remain exotic even after applying any number of positive twist moves.
Kouki Sato, Masaki Taniguchi
For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,q$, we obtain an infinite family of irreducible rational homology 3-spheres which are homology cobordant to the lens space $L(p,q)$ but cannot obtained by a knot surgery.
Hokuto Konno, Abhishek Mallick, Masaki Taniguchi
We initiate the study of exotic Dehn twists along 3-manifolds $\neq S^3$ inside $4$-manifolds, which produces the first known examples of exotic diffeomorphisms of contractible 4-manifolds, more generally of definite 4-manifolds, and exotic diffeomorphisms of 4-manifolds with $\neq S^3$ boundary that survive after one stabilization. We also construct the smallest closed 4-manifold known to support an exotic diffeomorphism. These exotic diffeomorphisms are the Dehn twists along certain Seifert fibered 3-manifolds. As a consequence, we get loops of diffeomorphisms of 3-manifolds that topologically extend to some 4-manifolds $X$ but not smoothly so, implying the non-surjectivity of $π_1(\mathrm{Diff}(X)) \to π_1(\mathrm{Homeo}(X))$. Our method uses $2$-parameter families Seiberg-Witten theory over $\mathbb{RP}^2$, while known methods to detect exotic diffeomorphisms used $1$-parameter families gauge-theoretic invariants. Using a similar strategy, we construct a new kind of exotic diffeomorphisms of 4-manifolds, given as commutators of diffeomorphisms.
Yoshihiro Aiura, Koji Sato, Hideaki Iwasawa, Yosuke Nakashima, Akihiro Ino, Masashi Arita, Kenya Shimada, Hirofumi Namatame, Masaki Taniguchi, Izumi Hase, Kiichi Miyazawa, Parasharam M. Shirage, Hiroshi Eisaki, Hijiri Kito, Akira Iyo
Photoemission spectroscopy with low-energy tunable photons on oxygen-deficient iron-based oxypnictide superconductors NdFeAsO0.85 (Tc=52K) reveals a distinct photon-energy dependence of the electronic structure near the Fermi level (EF). A clear shift of the leading-edge can be observed in the superconducting states with 9.5 eV photons, while a clear Fermi cutoff with little leading-edge shift can be observed with 6.0 eV photons. The results are indicative of the superconducting gap opening not on the hole-like ones around Gamma (0,0) point but on the electron-like sheets around M(pi,pi) point.
Ken-Ichi Aoki, Atsushi Horikoshi, Masaki Taniguchi, Haruhiko Terao
Aug 28, 2002·quant-ph·PDF We analyze quantum mechanical systems using the non-perturbative renormalization group (NPRG). The NPRG method enables us to calculate quantum corrections systematically and is very effective for studying non-perturbative dynamics. We start with anharmonic oscillators and proceed to asymmetric double well potentials, supersymmetric quantum mechanics and many particle systems.
Kenta Kuroda, Gaku Eguchi, Kaito Shirai, Masashi Shiraishi, Mao Ye, Koji Miyamoto, Taichi Okuda, Shigenori Ueda, Masashi Arita, Hirofumi Namatame, Masaki Taniguchi, Yoshifumi Ueda, Akio Kimura
The surfaces of three-dimensional topological insulators (TIs) characterized by a spin-helical Dirac fermion provide a fertile ground for realizing exotic phenomena as well as having potential for wide-ranging applications. To realize most of their special properties, the Dirac point (DP) is required to be located near the Fermi energy with a bulk insulating property while it is hardly achieved in most of the discovered TIs. It has been recently found that TlBiSe2 features an in-gap DP, where upper and lower parts of surface Dirac cone are both utilized. Nevertheless, investigations of the surface transport properties of this material are limited due to the lack of bulk insulating characteristics. Here, we present the first realization of bulk insulating property by tuning the composition of Tl1-xBi1+xSe2-d without introducing guest atoms that can bring the novel properties into the reality. This result promises to shed light on new exotic topological phenomena on the surface.
Sungkyung Kang, JungHwan Park, Masaki Taniguchi
We prove that every nontrivial cable of the figure-eight knot has infinite order in the smooth knot concordance group. Our main contribution is a uniform proof that applies to all $(2n,1)$-cables of the figure-eight knot. To this end, we introduce a family of concordance invariants $κ_R^{(k)}$, defined via $2^k$-fold branched covers and real Seiberg--Witten Floer $K$-theory. These invariants generalize the real $K$-theoretic Frøyshov invariant developed by Konno, Miyazawa, and Taniguchi.
Hokuto Konno, Abhishek Mallick, Masaki Taniguchi
We introduce a method to detect exotic surfaces without explicitly using a smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the construction. Our main tools are two versions of families (Seiberg-Witten) generalizations of Donaldson's diagonalization theorem, including a real and families version of the diagonalization. This leads to an example of a pair of exotically knotted $\mathbb{R}P^2$'s embedded in a closed 4-manifold whose complements are diffeomorphic, making it the first example of a non-orientable surface with this property. In particular, any invariant of a 4-manifold-surface pair (including invariants from real Seiberg-Witten theory such as Miyazawa's invariant) fails to detect such an exotic $\mathbb{R} P^2$. One consequence of our construction reveals that non-effective embeddings of corks can still be useful in pursuit of exotica. Precisely, starting with an embedding of a cork $C$ in certain a 4-manifold $X$ where the cork-twist does not change the diffeomorphism type of $X$, we give a construction that provides examples of exotically knotted spheres and $\mathbb{R}P^2$'s with diffeomorphic complements in $ C \# S^2 \times S^2 \subset X \# S^2 \times S^2$ or $C \# \mathbb{C}P^2 \subset X \# \mathbb{C}P^2 $. In another direction, we provide infinitely many exotically knotted embeddings of orientable surfaces, closed surface links, and 3-spheres with diffeomorphic complements in once stabilized corks, and show some of these surfaces survive arbitrarily many internal stabilizations. By combining similar methods with Gabai's 4D light-bulb theorem, we also exhibit arbitrarily large difference between algebraic and geometric intersections of certain family of 2-spheres, embedded in a 4-manifold.