Involve: A Journal of Mathematics

  • Involve
  • Volume 1, Number 2 (2008), 145-158.

On graphs for which every planar immersion lifts to a knotted spatial embedding

Amy DeCelles, Joel Foisy, Chad Versace, and Alice Wilson

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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.

Article information

Involve, Volume 1, Number 2 (2008), 145-158.

Received: 10 June 2007
Accepted: 1 December 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M15: Relations with graph theory [See also 05Cxx]

spatially embedded graph intrinsically linked intrinsically knotted regular projection


DeCelles, Amy; Foisy, Joel; Versace, Chad; Wilson, Alice. On graphs for which every planar immersion lifts to a knotted spatial embedding. Involve 1 (2008), no. 2, 145--158. doi:10.2140/involve.2008.1.145.

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