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Hoste’s Conjecture and Roots of the Alexander Polynomial

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

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Abstract

The Alexander polynomial remains one of the most fundamental invariants of knots and links in 3-space. Its topological understanding has led a long time ago to the insight of what (Laurent) polynomials occur as Alexander polynomials of arbitrary knots. Ironically, the question to characterize the Alexander polynomials of alternating knots turns out to be far more difficult, even although in general alternating knots are much better understood. J. Hoste, based on computer verification, made the following conjecture about 15 years ago: If z is a complex root of the Alexander polynomial of an alternating knot, then \(\mathfrak {R}e\,z > -1\). We discuss some results toward this conjecture, about 2-bridge (rational) knots or links, 3-braid alternating links, and Montesinos knots.

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Acknowledgements

The author thanks the referee and the Knots in Hellas organizers. It was a pleasure to attend the conference.

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Authors and Affiliations

  1. Gwangju Institute of Science and Technology, GIST College, 123 Cheomdan-gwagiro, Gwangju, 61005, Korea

    Alexander Stoimenov

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  1. Alexander Stoimenov
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Correspondence to Alexander Stoimenov .

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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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Stoimenov, A. (2019). Hoste’s Conjecture and Roots of the Alexander Polynomial. 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_9

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