My main research interests are in low dimensional topology and geometry, and gauge and Floer theoretic invariants for them. More specifically, my study focuses on the TQFT aspects of Ozsváth and Szabó's Heegaard Floer theory in dimensions (3+1). These include the study of knot concordance group, and the interplay between 4-manifolds and their boudary 3-manifolds

Research papers:

○ Irreducible 3-manifolds that cannot be obtanined by 0-surgery on a knot
with Matt Hedden, Min Hoon Kim, and Tom Mark, in preparation, 2018.

We give infinitely many examples of closed, orientable, irreducible 3-manifolds $M$ such that $b_1(M)=1$ and $\pi_1(M)$ has weight 1, but $M$ is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl and Wilton, and provides the first examples of irreducible manifolds with $b_1=1$ that are known not to be surgery on a knot in the 3-sphere. We further show that our examples are not even homology cobordant any manifold obtained by Dehn surgery along a knot in the 3-sphere, or any Seifert fibered 3-manifold.

○ Spherical 3-manifolds bounding rational homology balls
with Dong Heon Choe, in preparation, 2018.

We give a complete classification of the spherical 3-manifolds which smoothly bound rational homology 4-balls, and further determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology balls induced from Donaldson's diagonalization theorem and Heegaard Floer correction terms.

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Frøyshov, and Ozsvá th and Szabá. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.

○ An infinite-rank summand of knots with trivial Alexander polynomial
with Min Hoon Kim, to appear in Journal of Symplectic Geometry, arxiv:1604.04037, 2016.

We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $\Upsilon$ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute $\Upsilon$ of $(n,1)$-cable of the Whitehead double of the trefoil knot. For this computation of $\Upsilon$, we determine a sufficient condition for two satellite knots to have identical $\Upsilon$ for any pattern with nonzero winding number.

○ Some computations and applications of Heegaard Floer correction terms
MSU Thesis, 2014.

In this dissertation, we study some computations and applications of Heegaard Floer correction terms. In particular we explore the correction terms for the double covers of the three-sphere branched along the Whitehead doubles of knots. As a consequence we show that Whitehead double and iterated double of some classes of knots are independent in the smooth knot concordance group. We also compute the correction terms of non-trivial circle bundles over oriented surfaces and discuss how they can be applied to four-dimensional topology.

In this paper, we have studied a question for iterated Whitehead doubles and showed that for each $m>1$ the Whitehead double and iterated Whitehead double of $(2,2m+1)$ torus knots are not smoothly concordant and that indeed they generate a $\mathbb{Z}^2$ summand in the subgroup of the smooth knot concordance group generated by topologically slice knots. Our main tool is the Heegaard Floer correction term for the double cover of $S^3$ branched along a knot. We also present some sufficient conditions for general knots to have this independence property. Additionally, for some classes of knots including $(p,q)$ torus knots, we give an algorithmic formula for testing it in terms of its Alexander polynomial.