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An Invariant for Virtual Singular Links
by
Kelsey Renee Friesen
California State University, Fresno
A singular link is an immersion of a disjoint union of circles into three-dimensional space that admits only finitely many singularities that are all transverse double points. A singular link diagram is a projection of a singular link into a plane, and contains two types of crossings, namely classical crossings and singular crossings.
Virtual knot theory, introduced by Lou Kauffman in 1996, can be regarded as a projection of classical knot theory in thickened surfaces. We take one step further by studying virtual singular links, which can be thought as immersions of disjoint unions of circles into thickened surfaces. A virtual singular link diagram contains then three types of crossings: classical, singular, and virtual. Much as in the case of classical and virtual knot theory, when studying virtual singular links we seek for ways to tell them apart. An invariant for a virtual singular link is an object associated to it, which is independent on the link diagram, and may provide a powerful tool at distinguishing virtual singular links.
In this research, we employ a certain model for the sl(n) polynomial of classical links and extend it to a polynomial invariant for virtual singular links, which is defined as a state-sum formula. After this extension, we investigate how the polynomial behaves with respect to mirror images, disjoint unions, and compositions of virtual singular links.
Date received: February 15, 2016
Copyright © 2016 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbmk-03.