• Kyungpook Mathematical Journal
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• 제46권3호
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• pp.449-461
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• 2006

Generalizing twist moves of classical knots, we introduce $t(a_1,{\cdots},a_m)$-moves of virtual knots for an $m$-tuple ($a_1,{\cdots},a_m$) of nonzero integers. In [4], M. Goussarov, M. Polyak and O. Viro introduced finite type invariants of virtual knots and Gauss diagram formulae giving combinatorial presentations of finite type invariants. By using the Gauss diagram formulae for the finite type invariants of degree 2, we give a necessary condition for a virtual long knot K to be transformed to a virtual long knot K' by a finite sequence of $t(a_1,{\cdots},a_m)$-moves for an $m$-tuple ($a_1,{\cdots},a_m$) of nonzero integers with the same sign.

• Kyungpook Mathematical Journal
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• 제54권4호
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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.

• Kyungpook Mathematical Journal
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• 제58권1호
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• pp.183-202
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• 2018

${\Delta}-moves$ are closely related with a Vassiliev invariant of degree 2. For classical knots, M. Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single ${\Delta}-move$. The first author extended the Okada's result for virtual knots by using a Vassiliev invariant of virtual knots of type 2 which is induced from the Kauffman polynomial of a virtual knot. The arrow polynomial is a generalization of the Kauffman polynomial. We will generalize this result by using Vassiliev invariants of type 2 induced from the arrow polynomial of a virtual knot and give a lower bound for the number of ${\Delta}-moves$ transforming $K_1$ to $K_2$ if two virtual knots $K_1$ and $K_2$ are related by a finite sequence of ${\Delta}-moves$.