Link groups of Kishino knot stacks

TL;DR

This paper introduces a new class of links called stacks derived from virtual links, and demonstrates that their fundamental groups can distinguish Kishino knots from unknots and each other, also computing their Jones polynomials.

Contribution

It presents a novel method for constructing link groups from virtual links called stacks, which can detect nontrivial and nonclassical virtual knots like Kishino knots.

Findings

Groups distinguish all Kishino knots from the unknot

Groups differentiate Kishino knots from each other

Jones polynomials of Kishino knots are computed

Abstract

For any virtual link, a class of new links can be defined called stacks, in which copies of the virtual link are placed on top of one another. The resulting virtual link depends only on the virtual isotopy class of the original link, and the fundamental group of such a link may be used to detect whether the link is nontrivial and whether it is nonclassical in some cases. We show that the groups constructed using this method are sufficient to distinguish all the Kishino knots from the unknot and from one another, as well as calculating their Jones polynomials.