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My interests mainly lie in the areas of low-dimensional topology and geometry, with a focus on 3-manifolds. The topics overlap with questions from knot theory, computational topology, differential geometry, algebraic and quantum topology. Here is my Research Statement and Annotated Publication List.
 
The research below is/was supported by NSF DMS-2005496, NSF DMS-1664425, NSF DMS-1406588 individual research grants, by Institute of Advanced Study under DMS-1926686 grant (while I was a Von Neumann Fellow at IAS in 2020-2021), by Okinawa Institute of Science and Technology, and by an AWM grant. In 2022, NSF CAREER support is scheduled to start.
 
Publications and Preprints (all peer-reviewed; authors in alphabetical order; for works in preparation see my CV at the front page). 
 
Standard position for surfaces in link complements in arbitrary 3-manifolds, with J. Purcell, preprint, 27 pp., ArXiv
 
Random meander model for links, with N. Owad, preprint, 18pp., ArXiv,
 
NP-hard problems naturally arising in knot theory, with D. Koenig, Transactions of American Mathematical Society Ser. B 8 (2021), 420-441, ArXiv
 
Unlinking, splitting, and some other NP-hard problems in knot theory, with D. Koenig, Proceedings of Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM (2021), 1496–1507
 
Tangle decompositions of alternating link complements, with J. Hass and A. Thompson, llinois Journal of Mathematics 65 (2021), no. 3, 533–545, ArXiv
 
The number of Seifert surfaces of fixed genus is polynomial in the crossing number for an alternating link, with J. Hass and A. Thompson, Indiana University Mathematics Journal 70 (2021), no. 2, 525-534 , ArXiv
 
Simplicial volume of links from link diagrams, with O. Dasbach, Mathematical Proceedings of Cambridge Philosophical Society 166 (2019), no. 1, 75-81 , Arxiv
 
Determining isotopy classes of crossing arcs in alternating links, Asian Journal of Mathematics Vol. 22, No. 6 (2018), 1005-1024, ArXiv
 
The number of incompressible surfaces in an alternating link complement, with J. Hass and A. Thompson, International Mathematics Research Notices 6 (2017), 1611-1622, ArXiv
 
Intercusp parameters and the invariant trace field, with W. Neumann, Proceedings of the American Mathematical Society 14 (2016), No. 2, 887-896, ArXiv
 
A refined upper bound for the hyperbolic volume of alternating links and the colored Jones polynomial, with O. Dasbach, Mathematical Research Letters 22 (2015), No. 4, 1047-1060, ArXiv
 
Exact volume of hyperbolic 2-bridge links, Communications in Analysis and Geometry 22 (2014), No. 5, 881-896, ArXiv
 
An alternative approach to hyperbolic structures on link complements, with M. Thistlethwaite, Algebraic & Geometric Topology 14 (2014), 1307-1337, ArXiv
 
Hyperbolic Structures from Link Diagrams, Ph.D. Thesis, Unversity of Tennessee (2012), pdf
 
Decomposition Of Cellular Balleans, with I. V. Protasov, Topology Proceedings 36 (2010), 77-83, ArXiv
 
Asymptotic Rays, with O. Kuchaiev, International Journal of Pure Appl. Math. 56, no. 3 (2009), 353-358, ArXiv
 
 
Some Software
Implementation of the alternative method for computing hyperbolic structures of links. Windows version, currently written for alternating links with small regions only (written in C++).
Mathematica worksheet constructing the polynomial for the invariant trace field of a hyperbolic 2-bridge link.