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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799086

Digital Object Identifier
doi:10.2140/involve.2008.1.145

Mathematical Reviews number (MathSciNet)
MR2429655

Zentralblatt MATH identifier
1154.57003

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

Keywords
spatially embedded graph intrinsically linked intrinsically knotted regular projection

Citation

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. https://projecteuclid.org/euclid.involve/1513799086


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