My interests mainly lie in the areas of low-dimensional topology and geometry and knot theory. Some topics overlap with questions from computational topology, differential geomery or quantum topology.

My research is currently supported by the NSF grant DMS 1406588 and OIST funding.

Publications and Preprints

NP-hard problems naturally arising in knot theory, with D. Koenig, 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, ArXiv

Determining isotopy classes of crossing arcs in alternating links, to appear in Asian Journal of Mathematics, 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

Simplicial volume of links from link diagrams, with O. Dasbach, to appear in Mathematical Proceedings of Cambridge Philosophical Society, published online: 06 November 2017, pp. 1-7, 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

Hyperbolic structures from alternating link diagrams
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++).

Computing the invariant trace field from a link diagram, with no approximation involved
Mathematica worksheet constructing the polynomial for the invariant trace field of a hyperbolic 2-bridge link.

Materials from some previous talks

GEAR seminar at Rutgers University, Newark, 1/2017, "The number of surfaces of fixed genus in an alternating link complement" video

Redbud Triangulations conference, Oklahoma State University, 11/2016, "Isotopy classes of crossing arcs in alternating links" slides

The Thin Manifold conference, University of Iowa, 08/2014, "Intrinsic geometry and the invariant trace field of hyperbolic 3-manifolds" slides

Women Advancing Arizona Mathematics Colloquium, University of Arizona, 03/2014, "Hyperbolic links: diagrams, volume and the colored Jones polynomial"  slides

Low dimensional topology, knots, and orderable groups, Luminy-Marseille, France, 07/2013, "Exact volume of hyperbolic 2-bridge links" slides

GEAR Network Retreat, University of Illinois, Urbana-Champaign, 08/2012,"Hyperbolic structures from link diagrams"  video

Moab Topology Conference, Utah State University, 05/2012,"An alternative approach to hyperbolic structures on link complements"  slides

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