Research
My interests mainly lie in the areas of low-dimensional and computational topology and geometry, with a focus on 3-manifolds, links, and hyperbolic geometry. I’ve also been called a knot theorist, and it is perhaps correct.
The research below is/was supported by NSF CAREER grant (DMS-2142487), individual research grants NSF DMS-2005496, DMS-1664425, NSF DMS-1406588, Institute of Advanced Study under DMS-1926686 grant (while I was a Von Neumann Fellow at IAS), by Rutgers (as Rutgers Board of Trustees Research Fellowship for Scholarly Excellence), by Okinawa Institute of Science and Technology (while I was the Head of Geometry and Topology of Manifolds unit), and by an AWM grant.
Publications and Preprints (all peer-reviewed; authors in alphabetical order; for works in preparation see CV on the front page).
21. On complexity of alternating link equivalence, with T. Haideer, preprint, ArXiv
20. Polynomial bounds for surfaces in cusped 3-manifolds, with J. Purcell, preprint, ArXiv
19. Standard position for surfaces in link complements in arbitrary 3-manifolds, with J. Purcell, preprint, to appear in Algebraic and Geometric Topology, ArXiv
18. Random meander model for links, with N. Owad, preprint, Discrete & Computational Geometry, 72 (4), 1417-1436, 2024, ArXiv
17. NP-hard problems naturally arising in knot theory, with D. Koenig, Transactions of American Mathematical Society Ser. B 8 (2021), 420-441, ArXiv
16. 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
15. Tangle decompositions of alternating link complements, with J. Hass and A. Thompson, llinois Journal of Mathematics 65 (2021), no. 3, 533–545, ArXiv
14. 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
13. Simplicial volume of links from link diagrams, with O. Dasbach, Mathematical Proceedings of Cambridge Philosophical Society 166 (2019), no. 1, 75-81 , Arxiv
12. Determining isotopy classes of crossing arcs in alternating links, Asian Journal of Mathematics Vol. 22, No. 6 (2018), 1005-1024,ArXiv
11. 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
10. Intercusp parameters and the invariant trace field, with W. Neumann, Proceedings of the American Mathematical Society 14 (2016), No. 2, 887-896, ArXiv
9. 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
8. Exact volume of hyperbolic 2-bridge links, Communications in Analysis and Geometry 22 (2014), No. 5, 881-896, ArXiv
7. An alternative approach to hyperbolic structures on link complements, with M. Thistlethwaite, Algebraic & Geometric Topology 14 (2014), 1307-1337, ArXiv
6. Hyperbolic Structures from Link Diagrams, Ph.D. Thesis, Unversity of Tennessee (2012),pdf
5. Decomposition Of Cellular Balleans, with I. V. Protasov, Topology Proceedings 36 (2010), 77-83, ArXiv (Master’s degree paper)