arXiv subjects

Geunyoung Kim

Publications and source records attributed to Geunyoung Kim.

6 recordsLinked to original sources

arXiv2026

Bridge position of 3-manifolds embedded in the 5-sphere

We introduce and study bridge decompositions for 3-manifolds embedded in the 5-sphere. These generalize both the classical notion of bridge position for knots in the 3-sphere and the bridge trisections of surfaces in the 4-sphere due to Meier and Zupan. Our main technical tool is the multisections of 5-manifolds introduced by Aribi, Courte, Golla, and Moussard. We prove that every embedded 3-manifold admits such a decomposition; in particular, any such embedding is encoded by four trivial tangle diagrams. We also present a range of explicit examples, including $S^2$-spun knots and ribbon 3-knots.

math.GT↗

arXiv2026

An algorithm for Seifert surfaces in 3-manifolds via surgery presentations

The classical Seifert algorithm provides an explicit construction of a Seifert surface for any link in $S^3$. Alegria and Menasco extended this construction to integral homology $3$-spheres using Heegaard splittings. In this paper, we extend the Seifert algorithm to null-homologous links in arbitrary $3$-manifolds via surgery on framed links in $S^3$.

math.GT↗

arXiv2025

Heegaard diagrams for $5$-manifolds

We introduce a version of Heegaard diagrams for $5$-dimensional cobordisms with $2$- and $3$-handles, $5$-dimensional $3$-handlebodies, and closed $5$-manifolds. We show that every such smooth $5$-manifold can be represented by a Heegaard diagram, and that two Heegaard diagrams represent diffeomorphic $5$-manifolds if and only if they are related by certain moves. As an application, we construct Heegaard diagrams for $5$-dimensional cobordisms from the standard $4$-sphere to the Gluck twists along knotted $2$-spheres. This provides several statements equivalent to the Gluck twist being diffeomorphic to the standard $4$-sphere.

math.GT↗

arXiv2025

A note on some high-dimensional handlebodies

For $k \geq 0$ and $n \geq 2k+1$, we show that every $n$-dimensional $k$-handlebody is the product of a $2k$-dimensional $k$-handlebody and the standard $(n-2k)$-ball. For $k \geq 2$ and $n \geq 2k$, we introduce $(n,k)$-Kirby diagrams for some $n$-dimensional $k$-handlebodies, where $(4,2)$-Kirby diagrams correspond to the original Kirby diagrams for $4$-dimensional $2$-handlebodies.

math.GT↗

arXiv2023

Pants distances of knotted surfaces in 4-manifolds

We define a pants distance for knotted surfaces in 4-manifolds, which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the pair (X, F) admits a standard form (i.e., has simple topology). Furthermore, we calculate the exact values of our invariants for many new examples, such as the spun lens spaces. We provide a characterization of genus two quadrisections with distance at most six.

math.GT↗

arXiv2023

$n$-knots in $S^n\times S^2$ and contractible $(n+3)$-manifolds

In $1961$, Mazur constructed a contractible, compact, smooth $4$-manifold with boundary which is not homeomorphic to the standard $4$-ball, using a $0$-handle, a $1$-handle and a $2$-handle. In this paper, for any integer $n\geq2,$ we construct a contractible, compact, smooth $(n+3)$-manifold with boundary which is not homeomorphic to the standard $(n+3)$-ball, using a $0$-handle, an $n$-handle and an $(n+1)$-handle. The key step is the construction of an interesting knotted $n$-sphere in $S^n\times S^2$ generalizing the Mazur pattern.

math.GT↗