Intrinsic Linking and Knotting in Virtual Spatial Graphs

Details Intrinsic Linking and Knotting in Virtual Spatial Graphs Intrinsic Linking and Knotting in Virtual Spatial Graphs

2006-06-09

arxiv.org/abs/math/0606231v1

Alg. Geom. Top., Vol. 7, 2007, pp. 583-601

Mathematics

Geometric Topology

13 pages, 13 figures

Scientific paper

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2.

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