• East Asian mathematical journal
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• v.32 no.5
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• pp.767-774
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• 2016

Manturov inroduced the parity bracket polynomial by using a parity of virtual knots, which is an extension of Jones-Kauffman polynomial. We extend Manturov's result to virtual links, so that we obtain the parity bracket polynomial for virtual links and give some examples.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.469-474
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• 2011

The Kauffman bracket skein module $K_t$(M) of a 3-manifold M becomes an algebra for t = -1. We prove that this algebra has no non-trivial nilpotent elements for M being the exterior of the twist knot in 3-sphere and, therefore, it is isomorphic to the $SL_2(\mathbb{C})$-character ring of the fundamental group of M. Our proof is based on some properties of Chebyshev polynomials.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.1221-1248
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• 2022

In this paper we study the theory of knotoids and braidoids and the theory of pseudo knotoids and pseudo braidoids on the torus T. In particular, we introduce the notion of mixed knotoids in S², that generalizes the notion of mixed links in S³, and we present an isotopy theorem for mixed knotoids. We then generalize the Kauffman bracket polynomial, <; >, for mixed knotoids and we present a state sum formula for <; >. We also introduce the notion of mixed pseudo knotoids, that is, multi-knotoids on two components with some missing crossing information. More precisely, we present an isotopy theorem for mixed pseudo knotoids and we extend the Kauffman bracket polynomial for pseudo mixed knotoids. Finally, we introduce the theories of mixed braidoids and mixed pseudo braidoids as counterpart theories of mixed knotoids and mixed pseudo knotoids, respectively. With the use of the L-moves, that we also introduce here for mixed braidoid equivalence, we formulate and prove the analogue of the Alexander and the Markov theorems for mixed knotoids. We also formulate and prove the analogue of the Alexander theorem for mixed pseudo knotoids.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.639-653
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• 2014

In [9], Kauffman introduced virtual knot theory and generalized many classical knot invariants to virtual ones. For example, he extended the Jones polynomials $V_K(t)$ of classical links to the f-polynomials $f_K(A)$ of virtual links by using bracket polynomials. In [4], M. Goussarov, M. Polyak and O. Viro introduced finite type invariants of virtual knots. In this paper, we give a necessary condition for a virtual knot invariant to be of finite type by using $t(a_1,{\cdots},a_m)$-sequences of virtual knots. Then we show that the higher derivatives $f_K^{(n)}(a)$ of the f-polynomial $f_K(A)$ of a virtual knot K at any point a are not of finite type unless $n{\leq}1$ and a = 1.