Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection

Permanent URI for this collectionhttps://hdl.handle.net/11147/7148

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  • Article
    Biquandle Power Brackets of Oriented Links
    (Tubitak Scientific & Technological Research Council Turkey, 2025) Güğümcü, Neslihan; Nelson, Sam; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    In this paper, we introduce biquandle power brackets, an infinite family of invariants of oriented links containing the classical skein invariants and the quandle and biquandle 2-cocycle invariants as special cases. Biquandle power brackets are generalizations of biquandle brackets in which the values of Kauffman states also depend on the biquandle colors they admit. We provide example computations and discuss the relationship between these new invariants and the previous cases.
  • Article
    Citation - WoS: 7
    Citation - Scopus: 9
    Parity, Virtual Closure and Minimality of Knotoids
    (World Scientific Publishing, 2021) Güğümcü, Neslihan; Güğümcü, Neslihan; Kauffman, Louis H.; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    In this paper, we study parity in planar and spherical knotoids in relation to virtual knots. We introduce a planar version of the parity bracket polynomial for planar knotoids. We show that the virtual closure map (a map from the set of knotoids in S-2 to the set of virtual knots of genus at most one) is not surjective, by utilizing the surface bracket polynomial of virtual knots. We give specific examples of virtual knots that are not in the image of the virtual closure map. Turaev conjectured that minimal diagrams of knot-type knotoids have zero height. We prove this conjecture by using the results of Nikonov and Manturov induced by parities of virtual knots.
  • Article
    Citation - WoS: 2
    Citation - Scopus: 3
    Biquandle Brackets and Knotoids
    (World Scientific Publishing, 2021) Güğümcü, Neslihan; Güğümcü, Neslihan; Oyamaguchi, Natsumi; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper, we use biquandle brackets to enhance the biquandle counting matrix invariant defined by the first two authors in (N. Gügümcü and S. Nelson, Biquandle coloring invariants of knotoids, J. Knot Theory Ramif. 28(4) (2019) 1950029). We provide examples to illustrate the method of calculation and to show that the new invariants are stronger than the previous ones. As an application we show that the trace of the biquandle bracket matrix is an invariant of the virtual closure of a knotoid.