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Intrinsic linking and knotting in virtual spatial graphs

Thomas Fleming and Blake Mellor

Algebraic & Geometric Topology 7 (2007) 583–601

arXiv: math.GT/0606231

Abstract

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 nonterminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the virtual unknotting number of a knot, and show that any knot with nontrivial Jones polynomial has virtual unknotting number at least 2.

Keywords
spatial graph, intrinsically linked, intrinsically knotted, virtual knot
Mathematical Subject Classification 2000
Primary: 05C10
Secondary: 57M27
References
Publication
Received: 19 June 2006
Accepted: 13 March 2007
Published: 10 May 2007
Authors
Thomas Fleming
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093-0112
Blake Mellor
Mathematics Department
Loyola Marymount University
Los Angeles, CA 90045-2659
http://myweb.lmu.edu/bmellor