On Filamentations and Virtual Knots

Details On Filamentations and Virtual Knots On Filamentations and Virtual Knots

2001-12-19

arxiv.org/abs/math/0112206v2

Topology and its Applications. 134 (2003), 23-52.

Mathematics

Geometric Topology

Corrected/Updated version. 34 pages, 12 figures, 2 tables, latex document, xypic required to compile

Scientific paper

In this paper, we discuss filamentations on oriented chord diagrams. When a filamentation cannot be realized on an oriented chord diagram, then the corresponding flat virtual knot is non-trivial. If a flat knot diagram is non-trivial, then any virtual diagram whose shadow is the flat diagram must also be non-trivial. We introduce a class of flat diagrams for which no filamentations exist. This class gives the first example of an infinite set of virtual knots for which the Jones polynomial and the fundamental group both evaluate trivially. The related generalized Alexander polynomials for the virtual knots in this class turn out to be distinct, proving that the class is indeed infinite.

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