We describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle [KR], [FJK], the virtual quandle [Ma2], the ideas of quaternion biquandles by Roger Fenn and Andrew Bartholomew [BF], the concepts and properties of long virtual knots [Ma10], and other ideas in the interface between classical and virtual knot theory. In the present paper we present a new algebraic construction of virtual knot invariants, give various presentations of it, and study several examples. Several conjectures and unsolved problems are presented throughout the paper.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm188-0-6,
author = {Louis H. Kauffman and Vassily O. Manturov},
title = {Virtual biquandles},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {103-146},
zbl = {1088.57006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm188-0-6}
}
Louis H. Kauffman; Vassily O. Manturov. Virtual biquandles. Fundamenta Mathematicae, Tome 185 (2005) pp. 103-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm188-0-6/