arXiv subjects

Gheehyun Nahm

Publications and source records attributed to Gheehyun Nahm.

9 recordsLinked to original sources

arXiv2026

The Rasmussen s-invariant and exotic 4-manifolds

We give a short, analysis-free proof, inspired by Ren and Willis's, of the existence of exotic compact, orientable 4-manifolds. There are two distinguishing features of our proof. First, we avoid skein lasagna modules; we use Beliakova and Wehrli's generalization of Rasmussen's s-invariant to links in $S^{3}$ directly. Second, we reduce the complexity of the computations by choosing clever induction hypotheses in Stošić's induction scheme; this in particular allows us to avoid Ren and Willis's Comparison Lemma.

math.GT↗

arXiv2026

An irreducible real projective plane in the 4-sphere

We construct an irreducible embedded projective plane in $S^4$. This gives a counterexample to the Kinoshita conjecture and answers Problem 4.37 of the K3 problem list. Moreover, we answer both Questions (i) and (ii) of Problem 4.37: (i) the connected sum $R\# R$ is a Klein bottle in $S^4$ with extremal normal Euler number that does not admit an unknotted projective plane summand, and (ii) we show that our projective plane $R$ is irreducible by showing that the peripheral map $π_1 (\partial (S^4\setminus\mathring{N}(R)))\to π_1 (S^4 \setminus \mathring{N}(R))$ has kernel of order $2$.

math.GT↗

arXiv2026

Basepoints in Khovanov homology and nonorientable surfaces

We enhance the Khovanov TQFT using basepoint actions, over the field with two elements. Our enhanced Khovanov TQFT behaves similarly to gauge/Floer theoretic invariants of the double branched cover with opposite orientation: they both are invariant, in a certain sense, under taking the connected sum with the standard $\mathbb{RP}^{2}$ with Euler number -2, and they both vanish after taking the connected sum with the standard $\mathbb{RP}^{2}$ with Euler number 2. This invariance property answers a version of a question posed by Lipshitz and Sarkar. Furthermore, our construction establishes, as a special case, functoriality for the pointed Khovanov homology defined by Baldwin, Levine, and Sarkar.

math.GT↗

arXiv2026

Knotting and linking in 4 and 5 dimensions from barbell diffeomorphisms

In this paper, we construct infinitely many non-isotopic 3-knots in the 5-sphere, each of which has four critical points with respect to the standard height function of the 5-sphere. This contrasts with a theorem of Scharlemann which says that any 2-knot in the 4-sphere with four critical points is unknotted, and also provides infinitely many knotted solid tori in the 4-sphere and 5-ball, which resolves the last remaining case of the conjecture by Budney and Gabai on the existence of knotted handlebodies. We also construct various knotted and linked handlebodies, discs, and spheres in the 4-sphere, 5-ball, and 5-sphere, extending recent works of Hughes, Miller, and the first author, and a recent work of the authors. All of our examples are explicit and are constructed using barbell diffeomorphisms.

math.GT↗

arXiv2026

Brunnian links of 3-balls in the 4-sphere

For each integer $n\ge 2$, we construct infinitely many $n$-component Brunnian links of 3-balls in $S^4$. Our main tool is the third author's result on the existence of splitting spheres for the trivial two-component link of $2$-spheres in $S^{4}$; we also give a new proof of this.

math.GT↗

arXiv2026

Rational surgery exact triangles in Heegaard Floer homology

We construct a new family of surgery exact triangles in Heegaard Floer theory over the field with two elements. This family generalizes both Ozsváth and Szabó's $n$- and $1/n$-surgery exact triangles for positive integers $n$ and the author's recent 2-surgery exact triangle to all positive rational slopes. The construction reduces to a combinatorial problem that involves triangle and quadrilateral counting maps in a genus 1 Heegaard diagram. The main contribution of this paper is solving this combinatorial problem, which is particularly tricky for slopes $r\neq n,1/n$; one key idea is to use an involution that is closely related to the ${\rm Spin}^{c}$ conjugation symmetry.

math.GT↗

arXiv2025

Khovanov homology can distinguish exotic Mazur manifolds

In a recent breakthrough, Ren and Willis gave the first analysis-free proof of the existence of exotic compact, orientable 4-manifolds; their main tool is the Khovanov skein lasagna module defined by Morrison, Walker, and Wedrich. In this paper, we introduce a new, simple way of using Khovanov homology to distinguish certain exotic compact, orientable 4-manifolds; our new method does not depend on the skein lasagna module. As an application, we give the first analysis-free proof of the existence of exotic Mazur manifolds, i.e. compact, contractible 4-manifolds that have handle decompositions with a single 1-handle and a single 2-handle.

math.GT↗

arXiv2025

Spectral sequences in unoriented link Floer homology

In a previous work, we defined an unoriented skein exact triangle in unoriented link Floer homology. In this paper, we iterate a modified version of this exact triangle and obtain a spectral sequence from various versions of Khovanov homology to various versions of unoriented link Floer homology, over the field with two elements. In particular, for knots in $S^{3}$, we obtain a spectral sequence from the reduced Khovanov homology of the mirror of the knot to the knot Floer homology of the knot.

math.GT↗

arXiv2025

An unoriented skein exact triangle in unoriented link Floer homology

We define band maps in unoriented link Floer homology and show that they form an unoriented skein exact triangle. These band maps are similar to the band maps in equivariant Khovanov homology given by the Lee deformation. As a key tool, we use a Heegaard Floer analogue of Bhat's recent 2-surgery exact triangle in instanton Floer homology, which may be of independent interest. Unoriented knot Floer homology corresponds to $I^{\sharp}$ of the knot in our 2-surgery exact triangle.

math.GT↗