I am interested in using invariants of low dimensional manifolds coming from Heegaard Floer homology, to study knots, links and 3-manifolds.
Below are the research articles that I have written/been a part of so far:
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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.
Combinatorial Description of epsilon invariant for knots in S^3 (Joint with Hakan Doga) (Published in Journal of Knot Theory and its Ramifications)
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.
Rank Bounds in Link Floer Homology and Detection Results (Joint with Fraser Binns) (Published in Quantum Topology)
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).
Cable Links, Annuli and Sutured Floer homology (joint with Fraser Binns) (Published in Mathematical Research Letters)
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.
Unknotted curves on genus one Seifert surfaces of Whitehead doubles (joint with Colby Shaw, Veronica King, Bulent Tosun, Bruce Trace) (Published in Pacific Journal of Mathematics)
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.
Algorithms in 4-manifold topology (joint with Et. al, a collaborative work from the conference: 4-Manifolds and Algorithms in University of Regensburg, September 2024) (Submitted)
We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby--Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric.
In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.
This chapter briefly describes the motivation behind asking geometric questions regarding topological invariants of knots and links, and ends with an open question in that direction involving Heegaard Floer homology for links. This is based on the talk presented at ‘Knots, Quivers and Beyond’ conference at IIT Bombay, January 2025.
In Preparation:
FDTC of hyperbolic fibered slice knots (joint with Apratim Chakraborty)
Sutured Floer homology and detection of tangles (joint with Fraser Binns, Claudius Zibrowius)
A note on homology concordance group.
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