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Seungwon Kim

Publications and source records attributed to Seungwon Kim.

27 records · Page 2Linked to original sources
arXiv2021

Concordance invariants and the Turaev genus

We show that the differences between various concordance invariants of knots, including Rasmussen's $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the first example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar-Kofman.

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arXiv2020

Gluck twist and unknotting of satellite $2$-knots

In this paper, we show that the Gluck twist of certain satellite $2$-knots in a $4$-manifold do not change the diffeomorphism type in three different ways: one is directly from the definition of the satellite $2$-knot, and the other two are by finding an equivalent description of the satellite $2$-knot. Furthermore, using the new description, we gave infinite number of new examples of $2$-knots which are unknotted by connected summing a single standard real projective plane.

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arXiv2019

Trisections of surface complements and the Price twist

Given an $S\cong \mathbb{R}P^2$ smoothly embedded in a 4-manifold $X^4$ with Euler number 2 or -2, the Price twist is a surgery operation on $ν(S)$ yielding (up to) three different 4-manifolds: $X^4,τ_S(X^4),Σ_S(X^4)$. This is of particular interest when $X^4=S^4$, as then $Σ_S(X^4)$ is a homotopy 4-sphere which is not obviously diffeomorphic to $S^4$. In this paper, we show how to produce a trisection description of each Price twist on $S\subset X^4$ by producing a relative trisection of $X^4\setminusν(S)$. Moreover, we show how to produce a trisection description of general surface complements in 4-manifolds.

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arXiv2019

Isotopies of surfaces in 4-manifolds via banded unlink diagrams

In this paper, we study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected $4$-manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in $\mathbb{C}P^2$ (i.e. spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $\mathbb{C}P^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.

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arXiv2019

Turaev Surfaces

An expository article on Turaev surfaces written for "A Concise Encyclopedia of Knot Theory," to appear.

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arXiv2018

Immersed Möbius bands in knot complements and representatives of $\mathbb{Z}_2$-homology classes

We study the 3-dimensional immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number 1, and show that a knot has immersed crosscap number 1 if and only if it is a nonntrivial $(2p,q)$-torus or $(2p,q)$-cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the 4-dimensional crosscap number. We then use these constructions to find, for any $n\geq 2$, an oriented 3-manifold $Y_n$ and class $α_n \in H_2(Y_n;\mathbb{Z}_2)$ such that $α_n$ can be represented by an immersed $\mathbb{RP}^2$, but any embedded representative of $α_n$ has a component $S$ with $χ(S) \leq 1-n$.

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arXiv2016

A topological characterization of toroidally alternating knots

We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots which is different from Ito's recent characterization.

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arXiv2015

Link diagrams with low Turaev genus

We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple closed curves on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm.

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