{"id":358,"date":"2019-08-23T08:12:02","date_gmt":"2019-08-23T08:12:02","guid":{"rendered":"http:\/\/sites.rutgers.edu\/anastasiia-tsvietkova\/?page_id=358"},"modified":"2026-04-02T01:02:44","modified_gmt":"2026-04-02T01:02:44","slug":"research","status":"publish","type":"page","link":"https:\/\/sites.rutgers.edu\/anastasiia-tsvietkova\/research\/","title":{"rendered":"Research"},"content":{"rendered":"
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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.<\/p>\n<\/div>\n

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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), Rutgers (as Rutgers Board of Trustees Research Fellowship for Scholarly Excellence), Okinawa Institute of Science and Technology (while I was the Head of Geometry and Topology of Manifolds unit), and an AWM grant.<\/p>\n<\/div>\n

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Publications and Preprints <\/b>(all peer-reviewed; authors in alphabetical order).<\/p>\n<\/div>\n

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23. Polynomially many surfaces of fixed Euler characteristic in a hyperbolic 3-manifold<\/em>, with M. Lackenby<\/a>, preprint, ArXiv<\/a><\/p>\n<\/div>\n

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22. Geometric structures and PSL_2(C) representations of knot groups from knot diagrams<\/em>, with K. Petersen<\/a>, preprint, ArXiv<\/a><\/p>\n

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21. Polynomial algorithm for alternating link equivalence<\/em>, with T. Haideer<\/a>, preprint, ArXiv<\/a><\/p>\n

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20. Polynomial bounds for surfaces in cusped 3-manifolds<\/em>, with J. Purcell<\/a>, preprint, ArXiv<\/a><\/p>\n

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19. Standard position for surfaces in link complements in arbitrary 3-manifolds<\/em>, with J. Purcell<\/a>, Algebraic & Geometric Topology 26 (2026) 825-862, ArXiv<\/a><\/p>\n

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18. Random meander model for links<\/em>, with N. Owad<\/a>, preprint,\u00a0Discrete & Computational Geometry, 72 (4), 1417-1436, 2024, ArXiv<\/a><\/span><\/p>\n

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17. NP-hard problems naturally arising in knot theory<\/em>,\u00a0with\u00a0D. Koenig<\/a>, Transactions of American Mathematical Society Ser. B 8 (2021), 420-441, ArXiv<\/a><\/span><\/p>\n<\/div>\n

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16. Unlinking, splitting, and some other NP-hard problems in knot theory<\/em>, with D. Koenig<\/a>, Proceedings of Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM (2021), 1496–1507 (this is computer science conference version of the above)<\/p>\n<\/div>\n

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15. Tangle decompositions of alternating link complements,\u00a0<\/i> with\u00a0J. Hass<\/a> and A. Thompson<\/a><\/span>, llinois Journal of Mathematics\u00a065 (2021), no. 3, 533\u2013545, ArXiv<\/a><\/span><\/p>\n<\/div>\n

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14. The number of Seifert surfaces of fixed genus is polynomial in the crossing number for an alternating link,\u00a0<\/i>with\u00a0J. Hass<\/a>\u00a0and\u00a0A. Thompson<\/a>, Indiana University Mathematics Journal 70 (2021), no. 2, 525-534 , ArXiv<\/a><\/p>\n<\/div>\n

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13. Simplicial volume of links from link diagrams<\/i>, with\u00a0O. Dasbach<\/a>, Mathematical Proceedings of Cambridge Philosophical Society 166 (2019), no. 1, 75-81 ,\u00a0Arxiv<\/a><\/p>\n<\/div>\n

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12. Determining isotopy classes of crossing arcs in alternating links<\/i>, Asian Journal of Mathematics Vol. 22, No. 6 (2018), 1005-1024,ArXiv<\/a><\/span><\/p>\n<\/div>\n

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11. The number of incompressible surfaces in an alternating link complement,\u00a0<\/i>with\u00a0J. Hass<\/a>\u00a0and\u00a0A. Thompson<\/a>, International Mathematics Research Notices 6 (2017), 1611-1622,\u00a0ArXiv<\/a><\/p>\n<\/div>\n

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10. Intercusp parameters and the invariant trace field,\u00a0<\/i>with\u00a0W. Neumann<\/a>, Proceedings of the American Mathematical Society 14 (2016), No. 2, 887-896, ArXiv<\/a><\/p>\n<\/div>\n

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9. A refined upper bound for the hyperbolic volume of alternating links and the colored Jones polynomial,\u00a0<\/i>with\u00a0O. Dasbach<\/a>, Mathematical Research Letters 22 (2015), No. 4, 1047-1060,\u00a0ArXiv<\/a><\/p>\n<\/div>\n

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8. Exact volume of hyperbolic 2-bridge links<\/i>, Communications in Analysis and Geometry 22 (2014), No. 5, 881-896,\u00a0ArXiv<\/a><\/p>\n<\/div>\n

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7. An alternative approach to hyperbolic structures on link complements,\u00a0<\/i>with\u00a0M. Thistlethwaite<\/a>, Algebraic & Geometric Topology 14 (2014), 1307-1337,\u00a0ArXiv<\/a><\/p>\n<\/div>\n

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6. Hyperbolic Structures from Link Diagrams<\/i>, Ph.D. Thesis, Unversity of Tennessee (2012),pdf<\/a><\/i><\/p>\n<\/div>\n

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5. Decomposition Of Cellular Balleans,\u00a0<\/i>with\u00a0I. V. Protasov<\/a>, Topology Proceedings 36 (2010), 77-83,\u00a0ArXiv<\/a> (Master’s degree paper)<\/p>\n<\/div>\n

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4. Asymptotic Rays,\u00a0<\/i>with\u00a0O. Kuchaiev<\/a>, International Journal of Pure Appl. Math. 56, no. 3 (2009), 353-358,\u00a0ArXiv<\/a> (undergraduate paper)<\/div>\n
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Some Software <\/b>(more is listed in the CV)<\/div>\n
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3. Geometric structures from diagrams<\/a><\/i><\/div>\n
Implementation of the alternative method for computing equations for the canonical component of PSL(2,C)-representation variety of a knot, as well as the complete hyperbolic structure of a link. Written in Python. The method does not use any triangulation or polyhedral decomposition, and uses a link diagram instead. The code was written\/maintained jointly with Jaeyun Bae and<\/span>\u00a0<\/span>Dale Koenig, with contributions by others (see README for all names)<\/span>.<\/span><\/div>\n
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2. Hyperbolic structure from alternating link diagrams<\/a><\/i><\/div>\n
Implementation of the alternative method for computing the complete hyperbolic structure of links, written in C++. This version is for alternating links with small regions (2, 3, or 4 sides) only, but can be easily modified for larger regions. A more complete implementation is not this one, but the one is above. This older code is kept here in case anyone needs C++ version.<\/div>\n
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1. Computing the invariant trace field from a link diagram, with no approximation involved<\/a><\/i><\/div>\n
Mathematica worksheet constructing the polynomial for the invariant trace field of a hyperbolic 2-bridge link.<\/div>\n
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