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Citation - WoS: 7
Citation - Scopus: 8
Invariants of Multi-Linkoids
(Springer Basel Ag, 2023) Gabrovsek, Bostjan; Güğümcü, Neslihan
In this paper, we extend the definition of a knotoid to multilinkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the T-invariant in relation with generalized T-graphs.
Citation - WoS: 8
Citation - Scopus: 8
Invariants of Bonded Knotoids and Applications To Protein Folding
(MDPI, 2022) Güğümcü, Neslihan; Gabrovsek, Bostjan; Kauffman, Louis H.
In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid. We demonstrate that our invariants can be used for the analysis of the topological structure of proteins.
Citation - WoS: 7
Citation - Scopus: 9
Parity, Virtual Closure and Minimality of Knotoids
(World Scientific Publishing, 2021) Güğümcü, Neslihan; Kauffman, Louis H.
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.
Citation - WoS: 2
Citation - Scopus: 3
Biquandle Brackets and Knotoids
(World Scientific Publishing, 2021) Güğümcü, Neslihan; Nelson, Sam; Oyamaguchi, Natsumi
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.
Citation - WoS: 4
Citation - Scopus: 4
Quantum Invariants of Knotoids
(Springer, 2021) Güğümcü, Neslihan; Kauffman, Louis H.
In this paper, we construct quantum invariants for knotoid diagrams in R-2. The diagrams are arranged with respect to a given direction in the plane (Morse knotoids). A Morse knotoid diagram can be decomposed into basic elementary diagrams each of which is associated to a matrix that yields solutions of the quantum Yang-Baxter equation. We recover the bracket polynomial, and define the rotational bracket polynomial, the binary bracket polynomial, the Alexander polynomial, the generalized Alexander polynomial and an infinity of specializations of the Homflypt polynomial for Morse knotoids via quantum state sum models.

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