Vol. 1, No. 2, 2008

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ISSN: 1944-4184 (e-only)
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On graphs for which every planar immersion lifts to a knotted spatial embedding

Amy DeCelles, Joel Foisy, Chad Versace and Alice Wilson

Vol. 1 (2008), No. 2, 145–158
Abstract
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We call a graph G intrinsically linkable if there is a way to assign over/under information to any planar immersion of G such that the associated spatial embedding contains a pair of nonsplittably linked cycles. We define intrinsically knottable graphs analogously. We show there exist intrinsically linkable graphs that are not intrinsically linked. (Recall a graph is intrinsically linked if it contains a pair of nonsplittably linked cycles in every spatial embedding.) We also show there are intrinsically knottable graphs that are not intrinsically knotted. In addition, we demonstrate that the property of being intrinsically linkable (knottable) is not preserved by vertex expansion.

Keywords
spatially embedded graph, intrinsically linked, intrinsically knotted, regular projection
Mathematical Subject Classification 2000
Primary: 57M25, 57M15
Milestones
Received: 10 June 2007
Accepted: 1 December 2007
Published: 1 July 2008

Communicated by Ann Trenk
Authors
Amy DeCelles
Department of Mathematics
University of Minnesota
Minneapolis, MN 55455
United States
Joel Foisy
Department of Mathematics
SUNY Potsdam
Potsdam, NY 13676
United States
http://www2.potsdam.edu/foisyjs/
Chad Versace
Department of Mathematics
University of South Alabama
Mobile, AL 36688
United States
Alice Wilson
Department of Mathematics
SUNY Potsdam
Potsdam, NY 13676
United States