There exist no minimally knotted planar spatial graphs on the torus
We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.
All considered graphs are undirected finite graphs and we will work in the piecewise linear category. A graph embedding is an embedding of a graph in up to ambient isotopy and the corresponding spatial graph is the image of this embedding. A graph is planar if there exists an embedding . An embedding is trivial if is contained in a 2-sphere embedded in . Its image is a trivial spatial graph. A spatial graph is minimally knotted if is nontrivial but is trivial for every edge . Some authors call minimally knotted spatial graphs almost trivial, almost unknotted or Brunnian. In this paper, a nontrivial link is a nonsplit link with at least two components.
Previous research on minimally knotted spatial graphs has been undertaken: The first example of a minimally knotted spatial graph was an embedding of a handcuff graph given by Suzuki . Kawauchi , Wu  and Inaba and Soma  showed that every planar graph has a minimally knotted embedding. Ozawa and Tsutsumi  proved that minimally knotted embeddings of planar graphs are totally knotted. Especially minimally knotted -graphs have generated some interest. Kinoshita  gave the first example of a minimally knotted -graph (see Fig. 1) which Suzuki  generalised to give examples of minimally knotted -graphs for all . Closely related are ravels which are nontrivial embeddings of -graphs that contain no nontrivially knotted subgraph; this definition is equivalent to the one given by Farkas, Flapan and Sullivan . The concept of ravels has been introduced by Castle, Evans and Hyde  as local entanglements that are not caused by knots or links and may lead to new topological structures in coordination polymers. A ravel in a molecule has been synthesized by Lindoy et al . Castle, Evans and Hyde  conjectured the following:
Conjecture (Castle, Evans, Hyde ).
All nontrivial embeddings of planar graphs on the torus include a nontrivial knot or a nonsplit link.
With Theorem 1 we prove that their conjecture is true. With torus we refer to an embedded torus in the 3-sphere which may be nonstandardly embedded. A standardly embedded torus is a torus that bounds two solid tori in . A nonstandardly embedded torus still bounds a solid torus in by the Solid Torus Theorem .
Theorem 1 (Knots and links existence).
Let be a planar graph and be an embedding of with image .
If is contained in the torus and contains no nontrivial knot nor a nonsplit link,
then is trivial.
Since -graphs are planar, it follows from Theorem 1 that on the torus there exist no minimally knotted embeddings of -graphs with . This gives us the following:
Corollary 1 (Ravels do not embed on the torus).
Every nontrivial embedding of -graphs on the torus contains a knot.
We conclude by showing that all assumptions of Theorem 1 are necessary. Explicit ambient isotopies that transform spatial graphs which fulfil the assumptions of Theorem 1 into the plane , are given in . Another consequence of Theorem 1 that is stated in Remark 1 has been shown in  together with : Nontrivial 3-connected and simple planar spatial graphs that are embedded on a torus are chiral. A graph is simple if it contains no loops and no multi-edges. It is 3-connected if at least three vertices and their incident edges have to be deleted to decompose the graph or to reduce it to a single vertex. A spatial graph is chiral if it is not ambient isotopic to its mirror image.
2.1. Outline of the proof
The proof uses two theorems of Scharlemann, Thompson  and Ozawa, Tsutsumi . We assume that the spatial graph we consider is given by an embedding of a planar graph and furthermore that contains no nontrivially knotted or linked subgraph. We conclude that must be trivial. During the proof, we need the following two definitions: Definition 1. An embedding of a graph is primitive, if for each component of and any spanning tree of , the bouquet graph obtained from by contracting all edges of in is trivial. Definition 2. An embedding of a graph is free, if the fundamental group of is free. The argument of the proof is as follows: We start showing that the statement is true for nonstandardly embedded tori in Lemma 1. With Lemma 2 we argue that it is sufficient to consider connected graphs. Then we show in Lemma 3 that a bouquet graph on either contains a nontrivial knot or is trivial. Since any connected spatial graph on contracts to a bouquet graph on , it follows that is primitive if it contains no nontrivial knot. By Theorem 2 we know that the restriction is free for all connected subgraphs of . By Lemma 2 together with Theorem 3 we conclude that is trivial.
2.2. Preparations for the proof
Lemma 1 (Nonstandardly embedded torus).
Let be a torus that is not standardly embedded. Any spatial graph that is embedded on and that contains no nontrivial knot is trivial.
If the spatial graph contains a cycle that follows a longitude of the torus , this cycle is knotted since itself is knotted. Therefore, no such subgraph of can exist and we find a meridian of that has no intersection with . This shows that in embedded in the twice punctured sphere . Therefore, is trivial. ∎
It follows from Lemma 1 that the statement of Theorem 1 is true for nonstandardly embedded tori. Therefore, we only consider the standardly embedded torus from now on which saves us from considering different cases.
Lemma 2 (Connectivity Lemma).
The image of an embedding of a graph with connected components on the standard torus contains either a nonsplit link, or contains no nonsplit link and decomposes into disjoint components of which at least components are trivial.
Take any connected component of the embedding on the torus . The complement of in the torus (without considering the rest of the spatial graph ) is a collection of pieces that can be the punctured torus, discs and essential annuli without boundaries. (An essential annulus contains a simple closed curve that does not bound a disc in the torus.)
In the case that the complement of in includes the punctured torus, is trivial and splits from the other components.
If the complement of in is only a collection of discs, then all other components of lie in one of those discs and therefore are trivial and the graph is split. ( might or might not contain a nonsplit link.)
In the case that the complement of in includes an essential annulus , it is possible that other components of are embedded in this annulus. A component might be embedded in the annulus in two ways: Either the complement of in includes a punctured annulus and therefore is trivial and splits from the rest of the spatial graph . Or contains two annuli. The annulus has one type of an essential curve running inside it; is parallel to the boundary curves of . In the case that contains two annuli, a subgraph of must be deformable to be parallel to . If is a meridian or a prefered longitude of , both components and are split and trivial since the torus is a standard torus. If is neither a meridian nor a longitude of , and are nonsplittably linked. ∎
Lemma 3 (Bouquet Lemma).
The image of an embedding of a connected bouquet graph on the torus either contains a nontrivial knot or is trivial.
All cycles of a spatial bouquet graph on that contains no nontrivial knot are the unknot by assumption. The unknot on the torus can take the following forms:
loops that bound a disc in (trivial elements in ),
loops or alternatively loops,
Loops of type (1) do not contribute to nontriviality of .
If has loops of the types (1), (2) and (3) only, it is trivial.
If has loops of type (4), there are – beside the loops – only three types of loops simultaneously embeddable on the torus without self-intersections: and (respectively and ). This can easily be confirmed by applying the formula of Rolfsen’s exercise 2.7 : If two torus knots and intersect in one point transversally, then . Such a bouquet is trivial.
Theorem 2 (Ozawa and Tsutsumi’s freeness criterion ).
An embedding of a graph in is primitive if and only if the restriction is free for all connected subgraphs of .
Theorem 3 (Scharlemann and Thompson’s planarity criterion ).
An embedding of a graph is trivial if and only if
(a) is planar and
(b) for every subgraph , the restriction is free.
2.3. The proof
(of Theorem 1). It follows from Lemma 1 that the statement of Theorem 1 is true for nonstandardly embedded tori. Therefore, we assume that is embedded in the standard torus . By Lemma 2 (Connectivity Lemma) we can assume that is connected. Any connected spatial graph contracts to a spatial bouquet graph if a spanning tree is contracted in . If the spatial graph is embedded in a surface, edge contractions can be realised in the surface. It follows that a connected spatial graph which is embedded in the torus , contracts to a bouquet graph which also is embedded in if a spanning tree is contracted. Since contains no nontrivial knot by assumption, also contains no nontrivial knot. It follows from Lemma 3 that a bouquet graph that contains no nontrivial knot on the torus is trivial. Therefore, any bouquet graph which is obtained from by contracting all edges of in is trivial and is primitive by definition. By Theorem 2, the restriction is free for all connected subgraphs of . Then Lemma 2 ensures that the restriction is free for all subgraphs of since contains no nonsplit link by assumption. As is planar by assumption, it follows from Theorem 3 that is trivial. ∎
It has been shown in  together with  that every nontrivial embedding of a simple 3-connected spatial graph on the torus that contains a nontrivial knot or a nonsplit link is chiral. The following Remark 1 is therefore a consequence of Theorem 1.
Remark 1 (Chirality).
Nontrivial embeddings of simple 3-connected planar graphs in the torus are chiral.
2.4. All assumptions that have been made are necessary.
This can be seen by considering the following examples:
There exist nontrivial embeddedings on that contain neither a nontrivial knot nor a nonsplit link.
Those are embeddings of graphs which are not planar.
Examples: and embedded as shown left in Fig. 1.
There exist nontrivial embeddings of planar graphs that contain neither a nontrivial knot nor a nonsplit link.
Those are not embedded in the torus.
Examples: Kinoshita-theta curve (middle in Fig. 1) and every ravel.
There exist nontrivial embeddings of planar graphs on .
Examples: Spatial graphs that are subdivisions of nontrivial torus knots with vertices and edges (right in Fig. 1).
I thank Tom Coates, Erica Flapan, Youngsik Huh, Stephen Hyde, Danielle O’Donnol, Makoto Ozawa, Matt Rathbun and Kouki Taniyama for helpful comments and discussions. I also want to thank my PhD supervisor Dorothy Buck under whose supervision the research was undertaken. It was financially supported by the Roth studentship of Imperial College London mathematics department, the DAAD, the Evangelisches Studienwerk, the Doris Chen award, and by a JSPS grant awarded to Kouki Taniyama.
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