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On the Height of Knotoids

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Algebraic Modeling of Topological and Computational Structures and Applications (AlModTopCom 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 219))

Abstract

Knotoid diagrams are defined in analogy to open ended knot diagrams with two distinct endpoints that can be located in any region of the diagram. The height of a knotoid is the minimal crossing distance between the endpoints taken over all equivalent knotoid diagrams. We define two knotoid invariants; the affine index polynomial and the arrow polynomial that were originally defined as virtual knot invariants given in (Kauffman, J Knot Theory Ramif 21(3), 37, 2012) [6], (Kauffman, J Knot Theory Ramif 22(4), 30, 2013) [8], respectively, but here are described entirely in terms of knotoids in \(S^2\). We reprise here our results given in (Gügümcü, Kauffman, Eur J Combin 65C, 186–229, 2017) [3] that show that both polynomials give a lower bound for the height of knotoids.

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Acknowledgements

The first author would like to thank her supervisor Sofia Lambropoulou for several fruitful discussions and for her suggestion of the subject of knotoids for the author’s PhD study.

This research 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.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Neslihan Gügümcü

  2. Department of Mathematics, University of Illinois at Chicago, Chicago, IL, USA

    Louis H. Kauffman

Authors
  1. Neslihan Gügümcü
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  2. Louis H. Kauffman
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Corresponding author

Correspondence to Neslihan Gügümcü .

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Editors and Affiliations

  1. Department of Applied Mathematics, National Technical University of Athens, Athens, Greece

    Sofia Lambropoulou

  2. School of Chemical Engineering, National Technical University of Athens, Athens, Greece

    Doros Theodorou

  3. Department of Applied Mathematics, National Technical University of Athens, Athens, Greece

    Petros Stefaneas

  4. Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois, USA

    Louis H. Kauffman

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Gügümcü, N., Kauffman, L.H. (2017). On the Height of Knotoids. In: Lambropoulou, S., Theodorou, D., Stefaneas, P., Kauffman, L. (eds) Algebraic Modeling of Topological and Computational Structures and Applications. AlModTopCom 2015. Springer Proceedings in Mathematics & Statistics, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-68103-0_12

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