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JungHwan Park

Publications and source records attributed to JungHwan Park.

At least 19 recordsLinked to original sources
arXiv2026

$\widetilde{H}$-cobordisms, infinite cyclic covers, and real Seiberg--Witten theory

We study $\widetilde{H}$-cobordisms of distinguished homology handles, introduced by Kawauchi in 1976 using infinite cyclic covers. Despite the extensive development of gauge-theoretic and Floer-theoretic invariants since Kawauchi's work, none were previously known to distinguish smooth and topological $\widetilde{H}$-cobordism. In this paper, we construct asymptotic invariants of distinguished homology handles by applying real Seiberg--Witten theory to finite cyclic covers. Using these invariants, we show that the kernel of the natural map from the smooth $\widetilde{H}$-cobordism group to its topological counterpart contains a subgroup isomorphic to $\mathbb{Z}$. We also define spin versions of these groups and show that the kernel of the corresponding natural map contains a subgroup isomorphic to $\mathbb{Z}^\infty$.

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arXiv2026

Ribbon concordance and cabling

We study ribbon concordances to cable knots. We formulate a conjecture predicting that any nontrivial knot admitting a ribbon concordance to a (p,q)-cable must itself be a (p,q)-cable. We prove the conjecture when the knot admitting the ribbon concordance is already a cable of the same companion, is a torus knot, or has genus one. We also verify it for a broad class of target cables. The proofs use a minimum-height invariant defined from the immersed-curve formulation of knot Floer homology. This invariant obstructs ribbon concordances and implies that any nontrivial knot admitting a ribbon concordance to a fibered cable knot is prime. We also establish genus bounds for knots admitting ribbon concordances to cable knots.

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arXiv2026

Non-kinetic homotopy coherent actions on four-manifolds

We give the first example of a non-kinetic smooth homotopy coherent action of order two on a closed simply connected smooth four-manifold. This action is obtained by restricting a nontrivial smooth homotopy coherent action of the discrete circle group on a stabilized $K3$ surface. We also construct relatively non-kinetic smooth homotopy coherent extensions of boundary involutions over compact smooth $4$-manifolds. In addition, we exhibit boundary involutions that admit locally linear topological extensions but no smooth extensions over the same stabilized fillings. Nevertheless, we prove a Wall-type theorem showing that every free involution on a disjoint union of integral homology spheres extends smoothly over any simply connected smooth filling after sufficiently many stabilizations by $S^2\times S^2$.

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arXiv2026

Unknotting number, ribbon concordance, and singular instantons

We use equivariant singular instanton Floer theory with the Chern--Simons filtration to obstruct same-sign unknotting operations. We show that, for a large class of slice knots obtained through ribbon concordance, any unknotting sequence of null-homologous twists must contain both signs. The same method gives a $3$--manifold analogue, obstructing certain homology $3$--spheres from surgery on a knot and providing evidence for the monotonicity of the Dehn surgery number under ribbon homology cobordism.

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arXiv2026

Smooth Realizations of Line Configurations

We study the problem of realizing line configurations as collections of 2-spheres smoothly embedded in the complex projective plane. Building upon prior work by Ruberman and Starkston on topological realizations, we establish a stronger obstruction in the smooth category. Our proof relies on lattice-theoretic arguments based on Donaldson's diagonalization theorem.

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arXiv2026

Lifting Milnor Invariants for 3-Component Links

We define a sequence of integer-valued invariants $γ^k(L)$ for a $3$-component link $L$. We prove that the resulting $γ$-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift certain Milnor invariants of 3-component links. To establish this, we introduce an invariant $h(L)$, a $3$-component analogue of the Kojima--Yamasaki $η$-invariant, and show that it recovers the $γ$-invariants. As applications, we obtain a weak-cobordism classification when the distinguished component has trivial Alexander polynomial and characterize knots that bound continuously embedded disks in $B^4$ whose complements have fundamental group $\mathbb{Z}$.

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arXiv2026

Isotopy and equivalence of knots in 3-manifolds

We show that in a prime, closed, oriented 3-manifold M, equivalent knots are isotopic if and only if the orientation preserving mapping class group is trivial. In the case of irreducible, closed, oriented $3$-manifolds we show the more general fact that every orientation preserving homeomorphism which preserves free homotopy classes of loops is isotopic to the identity. In the case of $S^1\times S^2$, we give infinitely many examples of knots whose isotopy classes are changed by the Gluck twist.

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arXiv2026

Special alternating links of minimal unlinking number

For any link in the 3-sphere, there is a natural lower bound for the unlinking number in terms of the classical signature. We prove that if this lower bound is sharp for a special alternating link $L$, then the unlinking number of $L$ is necessarily realized by crossing changes in any alternating diagram for $L$. As an application, we compute new values of the unknotting numbers for some special alternating knots with crossing number 11 and 12.

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arXiv2026

Forbidden configurations and definite fillings of lens spaces

We study definite fillings of lens spaces. We classify the lens spaces $L(p,q)$ for which every smooth negative-definite filling $X$ satisfies \[ b_2(X)\ge b_2(X(p,q))-1, \] where $X(p,q)$ denotes the canonical negative-definite plumbing. The classification is given by 17 "forbidden configurations" that cannot appear as induced subgraphs of the canonical plumbing graph. More generally, we introduce a combinatorial framework that encodes the lattice embedding information coming from the dual plumbing of $X(p,q)$, and we prove that it is governed by a finite set of minimal forbidden configurations. We also discuss consequences for symplectic fillings of lens spaces and for smoothings of cyclic quotient singularities.

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arXiv2025

Ribbon knots and iterated cables of fibered knots

We define a knot to be $γ_0$-sharp if its Seifert genus is detected by the concordance invariant $γ_0$, which arises from the immersed curve formalism in bordered Heegaard Floer homology. We show that a connected sum of $γ_0$-sharp fibered knots is ribbon exactly when it is of the form $K \mathbin{\#} -K$. Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice--ribbon conjecture fails.

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arXiv2025

Smooth concordance of cables of the figure-eight knot

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.

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arXiv2025

Cables of the figure-eight knot via real Frøyshov invariants

We prove that the $(2n,1)$-cable of the figure-eight knot is not smoothly slice when $n$ is odd, by using the real Seiberg-Witten Frøyshov invariant of Konno-Miyazawa-Taniguchi. For the computation, we develop an $O(2)$-equivariant version of the lattice homotopy type, originally introduced by Dai-Sasahira-Stoffregen. This enables us to compute the real Seiberg-Witten Floer homotopy type for a certain class of knots. Additionally, we present some computations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots.

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arXiv2025

A satellite formula for real Seiberg-Witten Floer homotopy types

We establish a satellite formula for the real Seiberg-Witten Floer homotopy types of knots with odd patterns. Using this, we derive several applications to knot concordance theory. The satellite formula follows from a version of the excision theorem for real Floer homotopy types. Additionally, we show that the concordance invariants arising from real Seiberg-Witten theory depend only on the knot's zero-framed surgery.

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arXiv2024

Handle decomposition complexity and representation spaces

We prove that there are homology three-spheres that bound definite four-manifolds, but any such bounding four-manifold must be built out of many handles. The argument uses the homology cobordism invariant $Γ$ from instanton Floer homology.

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arXiv2024

A survey on embeddings of 3-manifolds in definite 4-manifolds

This article presents a survey on the topic of embedding 3-manifolds in definite 4-manifolds, emphasizing the latest progress in the field. We will focus on the significant role played by Donaldson's diagonalization theorem and the combinatorics of integral lattices in understanding these embeddings. Additionally, the article introduces a new result concerning the embedding of amphichiral lens spaces in negative-definite manifolds.

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arXiv2024

Definite fillings of lens spaces

This paper considers the problem of determining the smallest (as measured by the second Betti number) smooth negative-definite filling of a lens space. The main result is to classify those lens spaces for which the associated negative-definite canonical plumbing is minimal. The classification takes the form of a list of 10 "forbidden" subgraphs that cannot appear in the plumbing graph if the corresponding plumbed 4-manifold is minimal. We also show that whenever the plumbing is minimal any other negative-definite filling for the given lens space has the same intersection form up to addition of diagonal summands. Consequences regarding smooth embeddings of lens spaces in 4-manifolds are also discussed.

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arXiv2024

Exotic Dehn twists and homotopy coherent group actions

We consider the question of extending a smooth homotopy coherent finite cyclic group action on the boundary of a smooth 4-manifold to its interior. As a result, we prove that Dehn twists along any Seifert homology sphere, except the 3-sphere, on their simply connected positive-definite fillings are infinite order exotic.

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