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Kouki Sato

Publications and source records attributed to Kouki Sato.

At least 19 recordsLinked to original sources
arXiv2026

Structure and unique factorization in concordance groups of links

Donald and Owens introduced two link concordance groups with a marked component and showed that they contain the knot concordance group as a direct summand with infinitely generated complements. While not explicitly posed by Donald and Owens, the problem of determining the structure of these complements arises naturally from their work. In this paper, we completely resolve this problem by proving that both complements are isomorphic to $\mathbb{Z}^{\infty} \oplus (\mathbb{Z}/2\mathbb{Z})^{\infty}$. Moreover, we introduce a notion of prime element and establish a unique prime decomposition theorem. This yields a canonical normal form, providing a complete description of the group structure.

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arXiv2026

Cobordism maps in Khovanov homology and singular instanton homology I

Khovanov homology and singular instanton Floer homology are both functorial with respect to link cobordisms. Although the two homology groups are related by a spectral sequence, direct correspondence between the cobordism maps has not been rigorously established. In this paper, we define a cobordism map on the instanton cube complex as a filtered chain map, and prove that it recovers the cobordism maps both in Khovanov homology and singular instanton theory. In a sequel paper, we further extend this cobordism map to immersed cobordisms.

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arXiv2025

Cobordism maps in Khovanov homology and singular instanton homology II

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.

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arXiv2025

On the slice-torus invariant $q_M$ from $\mathbb{Z}_2$-equivariant Seiberg--Witten Floer cohomology

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 $\tilde{ss}_c$.

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arXiv2024

Counterexamples to Allen's conjectures

We show that the torus knots $T(2,5)$ and $T(2,9)$ bound smooth Möbius bands in the 4-ball whose double branched covers are negative definite, giving counterexamples to Conjectures 1.6 and 1.8 of Allen in [New York J. Math. 29 (2023) 1038-1059].

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arXiv2024

An unoriented analogue of slice-torus invariant

A slice-torus invariant is an $\mathbb{R}$-valued homomorphism on the knot concordance group whose value gives a lower bound for the 4-genus such that the equality holds for any positive torus knot. Such invariants have been discovered in many of knot homology theories, while it is known that any slice-torus invariant does not factor through the topological concordance group. In this paper, we introduce the notion of "unoriented slice-torus invariant", which can be regarded as the same as slice-torus invariant except for the condition about the orientability of surfaces. Then we show that the Ozsváth-Stipsicz-Szabó $\upsilon$-invariant, the Ballinger $t$-invariant and the Daemi-Scaduto $h_{\mathbb{Z}}$-invariant (shifted by a half of the knot signature) are unoriented slice-torus invariants. As an application, we give a new method for computing the above invariants, which is analogous to Livingston's method for computing slice-torus invariants. Moreover, we use the method to prove that any unoriented slice-torus invariant does not factor through the topological concordance group.

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arXiv2023

A family of slice-torus invariants from the divisibility of Lee classes

We give a family of slice-torus invariants $\tilde{ss}_c$, each defined from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements $c$ in any principal ideal domain $R$. For the special case $(R, c) = (F[H], H)$ where $F$ is any field, we prove that $\tilde{ss}_c$ coincides with the Rasmussen invariant $s^F$ over $F$. Compared with the unreduced invariants $ss_c$ defined by the first author in a previous paper, we prove that $ss_c = \tilde{ss}_c$ for $(R, c) = (F[H], H)$ and $(\mathbb{Z}, 2)$. However for $(R, c) = (\mathbb{Z}, 3)$, computational results show that $ss_3$ is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.

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arXiv2022

Instantons, special cycles, and knot concordance

We introduce a framework for defining concordance invariants of knots using equivariant singular instanton Floer theory with Chern-Simons filtration. It is demonstrated that many of the concordance invariants defined using instantons in recent years can be recovered from our framework. This relationship allows us to compute Kronheimer and Mrowka's $s^\sharp$-invariant and fractional ideal invariants for two-bridge knots, and more. In particular, we prove a quasi-additivity property of $s^\sharp$, answering a question of Gong. We also introduce invariants that are formally similar to the Heegaard Floer $τ$-invariant of Oszváth and Szabó and the $\varepsilon$-invariant of Hom. We provide evidence for a precise relationship between these latter two invariants and the $s^\sharp$-invariant. Some new topological applications that follow from our techniques are as follows. First, we produce a wide class of patterns whose induced satellite maps on the concordance group have the property that their images have infinite rank, giving a partial answer to a conjecture of Hedden and Pinzón-Caicedo. Second, we produce infinitely many two-bridge knots $K$ which are torsion in the algebraic concordance group and yet have the property that the set of positive $1/n$-surgeries on $K$ is a linearly independent set in the homology cobordism group. Finally, for a knot which is quasi-positive and not slice, we prove that any concordance from the knot admits an irreducible $SU(2)$-representation on the fundamental group of the concordance complement. While much of the paper focuses on constructions using singular instanton theory with the traceless meridional holonomy condition, we also develop an analogous framework for concordance invariants in the case of arbitrary holonomy parameters, and some applications are given in this setting.

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arXiv2022

Filtered instanton Floer homology and the homology cobordism group

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 $5_2$.

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arXiv2020

The bridge number of surface links and kei colorings

Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number $n$ for any integer $n \geq 4$. To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.

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arXiv2019

The $ν^+$-equivalence classes of genus one knots

The $ν^+$-equivalence is an equivalence relation on the knot concordance group. This relation can be seen as a certain stable equivalence on knot Floer complexes $CFK^{\infty}$, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this paper, we show that any genus one knot is $ν^+$-equivalent to one of the trefoil, its mirror and the unknot.

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arXiv2018

Rational homology 3-spheres and simply connected definite bounding

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.

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arXiv2017

On eigenvalues of double branched covers

For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, the dealternating number and the alternation number of knots, and we prove that Batson's bound for the non-orientable 4-genus gives an estimate of the value. In addition, we use the value to give a necessary condition for being quasi-alternating.

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arXiv2017

A full-twist inequality for the $ν^+$-invariant

Hom and Wu introduced a knot concordance invariant called $ν^+$, which dominates many concordance invariants derived from Heegaard Floer homology. In this paper, we give a full-twist inequality for $ν^+$. By using the inequality, we extend Wu's cabling formula for $ν^+$ (which is proved only for particular positive cables) to all cables in the form of an inequality. In addition, we also discuss $ν^+$-equivalence, which is an equivalence relation on the knot concordance group. We introduce a partial order on $ν^+$-equivalence classes, and study its relationship to full-twists.

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arXiv2016

Knots that are not slice either in positons or in negatons

An oriented compact 4-manifold $V$ with boundary $S^3$ is called a positon (resp. negaton) if its intersection form is positive definite (resp. negative definite) and it is simply connected. In this paper, we prove that there exist infinitely many knots which cannot bound null-homologous disks either in positons or in negatons. As a consequence, we find knots that cannot be unknotted either by only positive crossing changes or by only negative crossing changes.

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arXiv2015

Heegaard Floer correction terms of $(+1)$-surgeries along $(2,q)$-cablings

The Heegaard Floer correction term ($d$-invariant) is an invariant of rational homology 3-spheres equipped with a Spin$^c$ structure. In particular, the correction term of 1-surgeries along knots in $S^3$ is a ($2\mathbb{Z}$-valued) knot concordance invariant $d_1$. In this paper, we estimate $d_1$ for the $(2,q)$-cable of any knot $K$. This estimate does not depend on the knot type of $K$. If $K$ belongs to a certain class which contains all negative knots, then equality holds. As a corollary, we show that the relationship between $d_1$ and the Heegaard Floer $τ$-invariant is very weak in general.

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