Skip to main content
5 of 6
Added a footnote and an explanation about Theorem 2 of loc. cit. having an additional hypothesis that I here gloss over.
Peter Heinig
  • 6.1k
  • 1
  • 27
  • 47

It is somewhat surprising (to me) that what to me seems the simplest nontrivial example of theorems exactly fitting the question in the OP has not yet been mentioned in this thread: the embeddings of the Möbius ladders, which are finite simple undirected graphs, into $\mathbb{R}^3$.

This is an answer to

whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) embeddings of graphs into R^3 or S^3. If yes, can anyone show me any reference?

at least in the sense that in the very interesting article Erica Flapan: The Symmetries of the Möbius Ladder. Math. Ann. 283, 271-283 (1989), which I think, could be given a fruitful revival from a point of view of constructive mathematics (e.g., how much of Flapan's proofs/theorems can be done constructively?), the following was done, inter alia (there is more in Flapan's paper):

  • in Section 1 of loc. cit. a proof is given that there exists a graph $G$ and an automorphism $\alpha$ of $G$ as an abstract graph, such that there does not exist any embedding of (the geometric realization $\lvert G\rvert$ of ) $G$ into $\mathbb{R}^3$ such that $\alpha$ could be realized by an element of the group of leaving-$\lvert G\rvert$-invariant-as-a-point-set diffeomorphisms of $\mathbb{R}^3$. Note that in this rendition I rendered the author's "group of homeomorphisms of $G$ up to isotopy" into what to me evidently seems equivalent and clearer "automorphism of $G$ as an abstract graph". The example graph used by the author is $G:=$complete graph with six vertices.

  • in Section 2 of loc. cit., first a proof in classical logic is given that for any embedding $\eta\colon M_r\to S^3$ of the $r$-rung Möbius ladder $M_r$ into the $3$-sphere $\mathbb{S}^3$, any orientation-reversing diffeomorphism $\varphi\colon S^3\to S^3$ has the property that if it fixes1 $\mathrm{im}(\eta)$ as a pointset, then $\varphi$ does not fix at least $r-2$ of the $r$ rungs of $M_r$. This is, partly, an interesting counterfactural (since the author later gives a proof that for an odd number of rungs, such $\varphi$ are impossible): roughly, if there is a symmetry of the Euclidean-space-embedded Möbius ladder, then it must needs jumble almost all the ladder's rungs. This implication is then put to use to give a proof that for an odd number of $r$ungs, such a $\varphi$ is impossible. Roughly: you cannot reverse the orientation of an embedded odd-rung-number Möbius ladder graph by a Euclidean symmetry. The author then gives an example that for even number of rungs, such isometries do exist. I perceived this to be a result which is very relevant to the OP; it in particular is an interaction between a combinatorial property of the abstract graph (number of rungs) and a concept studied in knot theory: slightly vaguely, one could say: Flapan gave a proof that even-rung Möbius ladders are amphichiral, while odd-rung Möbius ladders are non-amphichiral. The smallest example of the latter ladder is the embedding represented by the illustration

enter image description here

on p. 272 of loc. cit. about which Flapan says that it is non-amphichiral, i.e. there does not exist any orientation-reversing self-diffeomorphism of $\mathbb{R}^3$ which whould fix it as a point set.

The author on p. 272 of loc. cit. writes

The property of topological achirality for graphs is analogous to the property of amphicheirality for knots.

  • in Section 3 of loc. cit., the focus shifts from (non-)existence of orientation-reversing Euclidean symmetries of embedded Möbius ladders, to properties of orientation-preserving such symmetries. The emphasis is on a difference between embeddings into $\mathbb{R}^3$ and embeddings into $S^3$.

This difference is illustrated by the author using the following illustration on p. 278 of loc. cit.

enter image description here

which is a an example of what the OP is asking for: this meant to represent a non-knot embedded into $S^3$, namely the abstract undirected simple graph $M_3$, the three-rung Möbius ladder.

Remarks.

  • I realized that there is an issue with the rendition I gave above: in loc. cit. the self-diffeomorphism of $S^3$ which is shown to be impossible is assumed to map what is called in loc.cit. the 'loop' $K$ of the geometrically Möbius ladder to itself, so strictly speaking, the above presentation claims that loc. cit. proved more than loc. cit. claims it proves. However, personally, I do not understand the hypothesis $h(K)=K$ in Theorem 2 of loc. cit. in the sense that to me it seems evident that the hypothesis $h(M_n)=M_n$ in Theorem 2 implies $h(K)=K$ (because of $h(M_n)=M_n$ $\Rightarrow$ the abstract graph-homomorphism $a$ defined by $h$ is a graph-automorphism of $M_n$ $\Rightarrow$ $a$ maps the abstract graph underlying the 'loop' $K$ to itself $\Rightarrow$ $h(K)=K$, the latter implication because $h$ is assumed to satisfy $h(M_n)=M_n$) so it seems it can simply be left out. This is what the footnote 1 is about.

1 Note that in loc. cit. there is a slightly stronger hypothesis than mere fixing the image of the embedding in its entirety. This hypothesis, to me, seems superfluous, so I think loc. cit. gives a proof of what is claimed in this thread.

Peter Heinig
  • 6.1k
  • 1
  • 27
  • 47