Research

Preprints /Publications

7. Cable links of uniformly thick knot types (joint with J. Etnyre, H. Min and T. Rodewald)

preprint : arxiv.org/abs/2507.03185 

Abstract: In this paper, we study Legendrian realizations of cable links of knot types that are uniformly thick but not Legendrian simple, extending prior work of Dalton, the second author, and Traynor. This leads to new phenomena, such as stabilized Legendrian links that are smoothly isotopic and component-wise Legendrian isotopic, but are not Legendrian isotopic. In our study, we establish new results for cable links whose cable slope is sufficiently negative. We will also show how to classify Legendrian knots in (most) negative cables of twist knots. This is done by introducing a new technique to the study of cables based on Legendrian surgeries.

6.   Algorithms in 4-manifold topology  (joint with S. Bastl, R. Burke, S, Dey, A. Durst, S. Friedl, D.Galvin, A. Garcia Rivas,  T. Hirsch, C. Hobohm, C.- S  Hsueh, M. Kegel,F. Kern, S. Lee, C Loeh, N. Manikandan,L. Mousseau, L. Munser, M. Pencovitch, P. Perras, M. Powell,J. P. Quintanilha,  L. Schambeck, D. Suchodoll, M. Tancer, A. Thiele, P. Truoel, M. Uschold, S. Vesela,M. Weiß, M. Wunsch-Rolshoven. )

     preprint: arxiv.org/abs/2411.08775 (under revision)

    Abstract: 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.

5.  Legendrian Hopf links in L(p,1).(joint with H. Geiges and S. Onaran)

   preprint: arxiv.org/abs/2409.02582  

Accepted for publication in The Quaterly Journal of Mathematics.

       Abstract:  We classify Legendrian realisations, up to coarse equivalence, of the Hopf link in the lens spaces L(p,1) with any contact structure. 

4.  Contact surgery numbers of Sigma(2,3,11) and L(4m+3,4) (joint with M. Kegel)

preprint: arXiv:2404.18177  (submitted)

Abstract:  We classify all contact structures with contact surgery number one on the Brieskorn sphere Sigma(2,3,11) with both orientations. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. We further prove similar results for some lens spaces: We classify all contact structures with contact surgery number one on lens spaces of the form L(4m+3,4). Along the way, we present an algorithm and a formula for computing the Euler class of a contact structure from a general rational contact surgery description and classify which rational surgeries along Legendrian unknots are tight and which ones are overtwisted. 

3.  Existence and construction of non-loose knots (joint with J. Etnyre, H. Min and A. Mukherjee).

preprint:  arXiv:2310.04908  

International Mathematics Research Notices, Volume 2025, Issue 14, July 2025

Abstract:   In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and discuss conditions under which non-looseness is preserved under cabling. 

2.  Transverse links, open books and overtwisted manifolds .  

preprint: arXiv:2108.07764 

New York J. Math. 29 (2023), 213-230

Abstract:  We prove that transverse links in any contact manifold (M,ξ) can be realized as a sub-binding of a compatible open book decomposition. We define the support genus of a transverse link and prove that the support genus of a transverse knot is zero if there is an overtwisted disk disjoint from it. Next, we find a relationship between the support genus of a transverse link and its Legendrian approximation.  

1.  Links in overtwisted contact manifolds.  

preprint: arXiv: 2011.12217 . 

Expo. Math 40 (2022), 231-248.

Abstract:  We prove that Legendrian and transverse links in overtwisted contact structures having overtwisted complements can be classified coarsely by their classical invariants. We further prove that any coarse equivalence class of loose links has support genus zero and constructed examples to show that the converse does not hold.