Finite Type Invariants and the Kauffman Bracket Polynomials of Virtual Knots

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 4,  2014, pp.639-653
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.4.639
Title & Authors
Finite Type Invariants and the Kauffman Bracket Polynomials of Virtual Knots
Jeong, Myeong-Ju; Park, Chan-Young; Yeo, Soon Tae;

Abstract
In [9], Kauffman introduced virtual knot theory and generalized many classical knot invariants to virtual ones. For example, he extended the Jones polynomials $\small{V_K(t)}$ of classical links to the f-polynomials $\small{f_K(A)}$ 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 $\small{t(a_1,{\cdots},a_m)}$-sequences of virtual knots. Then we show that the higher derivatives $\small{f_K^{(n)}(a)}$ of the f-polynomial $\small{f_K(A)}$ of a virtual knot K at any point a are not of finite type unless $\small{n{\leq}1}$ and a = 1.
Keywords
virtual knots;graphical finite type invariants;finite type invariants of virtual knots;
Language
English
Cited by
1.
On Gauss diagrams of periodic virtual knots, Journal of Knot Theory and Its Ramifications, 2015, 24, 10, 1540008
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