TL;DR
This paper develops a cobordism theory for virtual knots, providing new invariants, examples of virtual slice knots, and insights into virtual surface isotopy, advancing the understanding of virtual knot topology.
Contribution
It introduces a novel cobordism framework for virtual knots, including invariants, examples, and a generalized isotopy theory, expanding virtual knot classification methods.
Findings
Constructed non-trivial virtual slice knots
Determined four-ball genus of positive virtual knots
Formulated a virtual surface isotopy theory
Abstract
This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive virtual knots are given using the results of a companion paper by the author and Heather Dye and Aaron Kaestner. Problems related to band-passing are explained, and a theory of isotopy of virtual surfaces is formulated in terms of a generalization of the Yoshikawa moves.
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Materials and Mechanics · Computational Geometry and Mesh Generation