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Extending the Classical Skein

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

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 284))

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Abstract

We summarize the theory of a new skein invariant of classical links H[H] that generalizes the regular isotopy version of the Homflypt polynomial, H. The invariant H[H] is based on a procedure where we apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots and then we evaluate the resulting knots using the invariant H and inserting at the same time a new parameter. This procedure, remarkably, leads to a generalization of H but also to generalizations of other known skein invariants, such as the Kauffman polynomial. We discuss the different approaches to the link invariant H[H], the algebraic one related to its ambient isotopy equivalent invariant \(\Theta \), the skein-theoretic one and its reformulation into a summation of the generating invariant H on sublinks of a given link. We finally give examples illustrating the behaviour of the invariant H[H] and we discuss further research directions and possible application areas.

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Acknowledgements

Louis H. Kauffman’ s work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). Sofia Lambropoulou’s work has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation, MIS: 380154.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan St., Chicago, IL, 60607-7045, USA

    Louis H. Kauffman

  2. Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russia

    Louis H. Kauffman

  3. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, GR-157 80, Athens, Greece

    Sofia Lambropoulou

Authors
  1. Louis H. Kauffman
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  2. Sofia Lambropoulou
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Correspondence to Sofia Lambropoulou .

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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Kauffman, L.H., Lambropoulou, S. (2019). Extending the Classical Skein. 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_11

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