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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), by individual research grants NSF DMS-2005496, DMS-1664425, NSF DMS-1406588, by Institute of Advanced Study under DMS-1926686 grant (while I was a Von Neumann Fellow at IAS), 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).

On complexity of alternating link equivalence, with T. Haideer, preprint, ArXiv

Polynomial bounds for surfaces in cusped 3-manifolds, with J. Purcell, preprint, ArXiv

Standard position for surfaces in link complements in arbitrary 3-manifolds, with J. Purcell, preprint, to appear in Algebraic and Geometric Topology, ArXiv

Random meander model for links, with N. Owad, preprint, Discrete & Computational Geometry, 72 (4), 1417-1436, 2024, 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 (more is listed in the CV)

 

Implementation of the alternative method for computing equations for the canonical component of PSL(2,C)-character variety of a knot, as well as the complete hyperbolic structure of a link. It is 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, Mark Bell, Dale Koenig, Alex Lowen.

 

Implementation of the alternative method for computing hyperbolic structures of links, in written 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 (the first in the list). This older code is kept here in case anyone needs C++ version.

 

Mathematica worksheet constructing the polynomial for the invariant trace field of a hyperbolic 2-bridge link.