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If any embedding of your graph in 3-space has two cycles that are linked, then your graph is intrinsically linked (such as the Petersen graph). These graphs generalise non-planar graphs since for example any flat projection of these graphs will also be non-planar.

Non-planar graphs can also be characterised by Mac Lane's condition: there is a cycle basis with a cycle sharing an edge with more than 2 cycles — i.e. the graph has a non-trivial cycle involving a non-planar edge. This is informative since it gives a more constructive explanation of non-planarity, rather than just a forbidden minors characterisation.

I would like to understand better the way that linked graphs generalise non-planar graphs. Is there a condition analogous to Mac Lane's condition that characterises linked graphs?

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    $\begingroup$ Doesn't Mac Lane's condition work the other way? I.e., a graph is non-planar if and only if all of its cycles bases have an edge contained in more than 2 cycles $\endgroup$
    – Arnaud
    Commented Jul 12, 2021 at 15:30
  • $\begingroup$ Graphs are linklessly embeddable if and only if their $\mu$-invariant of Colin de Verdi`ere is at most $4$. See the pertaining Wikipedia page. $\endgroup$
    – Roland Bacher
    Commented Jul 12, 2021 at 15:36
  • $\begingroup$ Geometric conditions and reformulations were given by H. van der Holst (see the Wikipedia on $\mu$ for exact refences) if I am not mistaken, The $\mu$-invariant has a combinatorial description (found by L. Laszlo and Schrijver) if I remember correctly. $\endgroup$
    – Roland Bacher
    Commented Jul 12, 2021 at 15:51
  • $\begingroup$ Thanks @RolandBacher yes I think the invariant introduced in Holst09 (related to the $\mu$-invariant) is closest to what I'm looking for - it seems to generalise the Hanani–Tutte rather Mac Lane characterisation of planarity. $\endgroup$
    – ben macintosh
    Commented Jul 14, 2021 at 7:26

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