I am interested in using invariants of low dimensional manifolds coming from Heegaard Floer homology, to study knots, links and 3-manifolds. 

 Below are my research so far:

As the main result of this paper, we prove that cables knot are not Floer thin and hence not quasi-alternating. This gives a partial answer to a conjecture in knot theory that satellite knots are not quasi-alternating. We use bordered Floer homology of Lipshitz-Ozsv ́ath-Thurston to find two non-zero elements in the knot Floer homology of cable knots which violate the Floer thinness condition. As a corollary to our calculation, we also show that a bigger family of satellite knots is indeed not Floer thin and hence not quasi-alternating. 

In this paper, we give a combinatorial description of the concordance invariant epsilon defined by Jen Hom in knot Floer homology setting and prove some properties of this invariant using only Grid Homology techniques and recover some of the results of Hom about iterated cables of torus knots. 

Viewing the BRAID invariant as a generator of link Floer homology we generalise work of Baldwin-Vela-Vick to obtain rank bounds on the next to top grading of knot Floer homology. These allow us to classify links with knot Floer homology of rank at most eight, and prove a variant of a classification of links with Khovanov homology of low rank due to Xie-Zhang. In another direction we use a variant of Ozsv'ath-Szab'o classification of E_2 collapsed bi-filtered chain complexes to show that knot Floer homology detects T(2,8) and T(2,10). Combining these techniques with the spectral sequences of Batson-Seed, Dowlin, and Lee we can show that Khovanov homology likewise detects T(2,8) and T(2,10).

We apply sutured Floer homology techniques to study the knot and link Floer homologies of various links with annuli embedded in their exteriors. Our main results include, for large n, characterizations of links with the same link Floer homology as (m,mn) cables of L-space knots, or the same knot Floer homology as (2,2n)-cables of L-space knots. These yield some new link detection results.

  We consider homologically essential simple closed curves on Seifert surfaces of genus one knots in 3-sphere, and in particular those that are unknotted or slice in 3-sphere. We completely characterize all such curves for most twist knots: they are either positive or negative braid closures; moreover, we determine exactly which of those are unknotted. A surprising consequence of our work is that the figure eight knot admits infinitely many unknotted essential curves up to isotopy on its genus one Seifert surface, and those curves are enumerated by Fibonacci numbers. On the other hand, we prove that many twist knots admit homologically essential curves that cannot be positive or negative braid closures. Indeed, among those curves, we exhibit an example of a slice but not unknotted homologically essential simple closed curve. We further investigate our study of unknotted essential curves for arbitrary Whitehead doubles of non-trivial knots, and obtain that there is a precisely one unknotted essential simple closed curve in the interior of the doubles' standard genus one Seifert surface. As a consequence of all these we obtain many new examples of 3-manifolds that bound contractible 4-manifolds.

Martin showed that link Floer homology detects braid axes. In this paper we extend this result to give a topological characterisation of links which are almost braided from the point of view of link Floer homology. The result is inspired by work of Baldwin-Sivek and Li-Ye on nearly fibered knots. Applications include that Khovanov homology detects the Whitehead link and L7n2, as well as infinite families of detection results for link Floer homology and annular Khovanov homology. 

In Preparation: