Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWrithe polynomials and shell moves for virtual knots and links
99
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links.
rate research
Read More
We give a new interpretation of the Alexander polynomial $Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffmans affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.
Oikawa defined an unknotting operation on virtual knots, called a CF-move, and gave a classification of 2-component virtual links up to CF-moves by the virtual linking number and his $n$-invariant. In particular, it was proved that a CF-move characterizes the information contained in the virtual linking number for 2-component odd virtual links. In this paper, we extend this result by classifying odd virtual links and almost odd virtual links with arbitrary number of components up to CF-moves, using the virtual linking number. Moreover, we extend Oikawas $n$-invariant and introduce two invariants for 3-component even virtual links. Using these invariants together with the virtual linking number, we classify 3-component even virtual links up to CF-moves. As a result, a classification of 3-component virtual links up to CF-moves is provided.
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
The Jones polynomial $V_{L}(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $nge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_{n}$-equivalent is divisible by $left(t-1right)^{n}left(t^{2}+t+1right)left(t^{2}+1right)$, and (2) there exists a pair of two oriented knots which are $C_{n}$-equivalent such that the difference of the Jones polynomials for them equals $left(t-1right)^{n}left(t^{2}+t+1right)left(t^{2}+1right)$.
Let $n$ be a positive integer. M. K. Dabkowski and J. H. Przytycki introduced the $n$th Burnside group of links which is preserved by $n$-moves, and proved that for any odd prime $p$ there exist links which are not equivalent to trivial links up to $p$-moves by using their $p$th Burnside groups. This gives counterexamples for the Montesinos-Nakanishi $3$-move conjecture. In general, it is hard to distinguish $p$th Burnside groups of a given link and a trivial link. We give a necessary condition for which $p$th Burnside groups are isomorphic to those of trivial links. The necessary condition gives us an efficient way to distinguish $p$th Burnside groups of a given link and a trivial link. As an application, we show that there exist links, each of which is not equivalent to a trivial link up to $p$-moves for any odd prime $p$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
