arXiv subjects

Sungkyung Kang

Publications and source records attributed to Sungkyung Kang.

26 records · Page 2Linked to original sources

arXiv2021

Stabilization and satellite construction of doubly slice links

A 2-component oriented link in $S^3$ is called weakly doubly slice if it is a cross-section of an unknotted sphere in $S^4$, and strongly doubly slice if it is a cross-section of a 2-component trivial spherical link in $S^4$. We give the first example of 2-component boundary links which are weakly doubly slice but not strongly doubly slice. We also introduce a new invariant $g_{st}$ of homotopically trivial links that measures the failure of a link from being strongly doubly slice and that bounds the doubly slice genus $g_{ds}$ from below. Our examples have arbitrarily large doubly slice genus but satisfy $g_{st}=1$. We also prove that the Conway-Orson signature lower bound on $g_{ds}$ is actually a lower bound on $g_{st}$.

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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

Ribbon knots, cabling, and handle decompositions

The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the (p,1)-cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number p. Our main tools are Juhász-Miller-Zemke's bound on fusion number coming from the torsion order of knot Floer homology and Hanselman-Watson's cabling formula for immersed curves.

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arXiv2019

Link homology theories and ribbon concordances

It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.

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arXiv2019

Connected Floer homology of covering involutions

Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.

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arXiv2018

$\mathbb{Z}_{2}$-equivariant Heegaard Floer cohomology of knots in $S^{3}$ as a strong Heegaard invariant

The $\mathbb{Z}_{2}$-equivariant Heegaard Floer cohomlogy $\widehat{HF}_{\mathbb{Z}_{2}}(Σ(K))$ of a knot $K$ in $S^{3}$, constructed by Hendricks, Lipshitz, and Sarkar, is an isotopy invariant which is defined using bridge diagrams of $K$ drawn on a sphere. We prove that $\widehat{HF}_{\mathbb{Z}_{2}}(Σ(K))$ can be computed from knot Heegaard diagrams of $K$ and show that it is a strong Heegaard invariant. As a topolocial application, we construct a transverse knot invariant $\hat{\mathcal{T}}_{\mathbb{Z}_{2}}(K)$ as an element of $\widehat{HFK}_{\mathbb{Z}_{2}}(Σ(K),K)$, which is a refinement of $\widehat{HF}_{\mathbb{Z}_{2}}(Σ(K))$, and show that it is a refinement of both the LOSS invariant $\hat{\mathcal{T}}(K)$ and the $\mathbb{Z}_{2}$-equivariant contact class $c_{\mathbb{Z}_{2}}(ξ_{K})$.

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arXiv2018

A transverse knot invariant from Z2-equivariant Heegaard Floer cohomology

We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ along based knots, and then prove that our transverse link invariant $c_{\mathbb{Z}_{2}}(ξ_{K})$ is an well-defined element which is always nonvanishing and functorial under certain classes of symplectic cobordisms, and describe its behavior under negative stabilization. It follows that we can use properties of $c_{\mathbb{Z}_{2}}(ξ_{K})$ to give a condition on transverse knots K which implies the vanishing/nonvanishing of the contact class $c(ξ_{K})$.

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arXiv2018

Spectral order for contact manifolds with convex boundary

We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a $2π$ Giroux torsion domain has order at most two, hence any contact structure with positive Giroux torsion has order at most two (and, in particular, a vanishing contact invariant).

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