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

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





Publications

  • 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, 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

Selected 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.


Materials from some of the 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