On Vassiliev Invariants of Virtual Knots
Wout Moltmaker, Louis H. Kauffman
TL;DR
This paper explores Vassiliev invariants for virtual knots, extending quantum invariants to rotational virtual knots, and introduces new tools like chord diagrams and weight systems to enhance invariant detection.
Contribution
It develops the theory of Vassiliev invariants for rotational virtual knots, including new definitions and examples of weight systems, and discusses extended quantum invariants.
Findings
Defined chord diagrams and weight systems for rotational virtual knots
Provided examples of Lie algebra weight systems for these knots
Discussed extended quantum invariants that capture additional information
Abstract
We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.
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Taxonomy
TopicsGeometric and Algebraic Topology · Logic, programming, and type systems · Homotopy and Cohomology in Algebraic Topology