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A Survey of Hyperbolic Knot Theory

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Knots, Low-Dimensional Topology and Applications (KNOTS16 2016)

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Abstract

We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.

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Notes

  1. 1.

    WYSIWYG stands for “what you see is what you get”.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    David Futer

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Efstratia Kalfagianni

  3. School of Mathematical Sciences, Monash University, Clayton, VIC, 3800, Australia

    Jessica S. Purcell

Authors
  1. David Futer
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  2. Efstratia Kalfagianni
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  3. Jessica S. Purcell
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Corresponding author

Correspondence to Efstratia Kalfagianni .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Williams College, Williamstown, MA, USA

    Colin C. Adams

  2. Department of Mathematics, University of Texas at Austin, Austin, TX, USA

    Cameron McA. Gordon

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Vaughan F.R. Jones

  4. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    Louis H. Kauffman

  5. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

    Sofia Lambropoulou

  6. Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, USA

    Kenneth C. Millett

  7. Department of Mathematics, Columbian College of Arts & Sciences, George Washington University, Washington, DC, USA

    Jozef H. Przytycki

  8. Department of Mathematics and Applications, University of Milano-Bicocca, Milano, Italy

    Renzo Ricca

  9. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Radmila Sazdanovic

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Futer, D., Kalfagianni, E., Purcell, J.S. (2019). A Survey of Hyperbolic Knot Theory. In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_1

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