On chirality of toroidal embeddings of polyhedral graphs
OON CHIRALITY OF TOROIDAL EMBEDDINGS OF POLYHEDRAL GRAPHS
SENJA BARTHELA
BSTRACT . We investigate properties of spatial graphs on the standard torus. It is known that nontrivialembeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building onthis and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivialknot, the statement was shown by Castle et al [5]. We give an alternative proof using minors instead of the Eulercharacteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chiralityof Hopf ladders with at least three rungs, thus generalising a theorem of Simon [6]. topological graphs; knots and links; chirality; topology and chemistry; templating on a toroidal substrate1. I
NTRODUCTION
The collaboration between mathematicians working in knot theory and topological graph theory andchemists working in stereochemistry has been very fruitful so far ([7]-[11]). The spatial arrangement ofa molecule can be modeled by a spatial graph G , which is the image of an embedding f : G → R ofan abstract graph G into R up to ambient isotopies ; i.e., bending, stretching and shrinking of G withoutself-intersections is allowed as long as no edge is collapsed. The value of modeling a molecular structure bya spatial graph lies in the fact that topological properties of the spatial graph are inherited by the molecule.For example, topological chirality implies chemical chirality. A spatial graph is (topologically) chiral ifit is not ambient isotopic to its mirror image. It is achiral otherwise; this is equivalent to the existence ofan orientation-reversing homeomorphism of R that maps the spatial graph onto itself. A molecule with anunderlying chiral graph is automatically a chiral molecule.Castle, Evans and Hyde [5] proved that polyhedral toroidal molecules which contain a nontrivial knot arechiral. A polyhedral molecule has an underlying graph which is planar, 3-connected and simple. A graphis planar if there exists an embedding of the graph in the sphere S (or equivalently in the plane R ). Suchan embedding is a trivial embedding and its image is a trivial spatial graph . A graph is n -connected ifat least n vertices and their incident edges have to be removed to decompose the graph or to reduce it to asingle vertex. A graph is simple if it has neither multiple edges between a pair of vertices nor loops froma vertex to itself. Nontrivial spatial graphs (as well as their corresponding molecules) which embed in thestandard torus are called toroidal .The argument given in [5] to show the chirality of polyhedral toroidal molecules which contain a nonsplitlink depends partly on a theorem of Simon [6] whose conditions unfortunately are not satisfied in [5].Simon’s theorem states that the Hopf ladder with at least three rungs is chiral, assuming that sides are takento sides. The Hopf ladder H n is the spatial graph which is obtained from a Hopf link by adding n pairsof vertices ( v , v ) , . . . , ( v n , v n ), where the vertices v , . . . , v n lie on one link component and the vertices v , . . . , v n lie on the other, and by adding n edges called rungs e , . . . , e n , where e i has endpoints v i , v i sothat the rungs do not introduce any crossings in a diagram of the spatial graph as illustrated for H and H in Fig. 1. We generalise Simon’s theorem with the following proposition: Proposition 1.1.
The Hopf ladder H n with n rungs is achiral if ≤ n ≤ , chiral if ≤ n . This allows us to complete the proof in [5] using a different method (minors instead of the Euler charac-teristic) and to obtain the following result:
Theorem 1.2 (Chirality) . Nontrivial simple 3-connected planar graphs which are embedded in the standard torus T are chiral. a r X i v : . [ m a t h . G T ] A p r or the proof we rely on the fact that all planar toroidal spatial graphs contain a nontrivial knot or anonsplit link: Theorem 1.3 (Existence of knots and links [1],[2]) . Let G be an planar graph and f : G → R be a nontrivial embedding of G with image G . If G is containedin the torus T , it contains a subgraph which is a nontrivial knot or a nonsplit link.
2. T
HE PROOFS
The first part of this section contains the outline of the proof of Theorem 1.2 and preparations for it, inparticular the generalisation of Simon’s theorem on the chirality of Hopf ladders. The second subsectionproves Theorem 1.2.2.1.
Outline of the proof and chirality of Hopf ladders.
The proof of Theorem 1.2 uses Theorem 1.3,Proposition 1.1 and the following three statements.
Theorem 2.1 (Chirality of torus knots and torus links [3],[4]) . Torus knots and torus links with at least three crossings are chiral.
Remark 2.2 (A necessary condition for a spatial graph to be achiral) . Let K ( G ) be the set of all knots andlinks up to ambient isotopy contained as subgraphs in the spatial graph G . If G is achiral, every topologicalchiral element in the set K ( G ) can be deformed into the mirror image of an element in the set K ( G ) . Remark 2.2 says that if an achiral spatial graph G contains a chiral spatial subgraph K , it must alsocontain the mirror image K (cid:48) of K . Note that K and K (cid:48) need not be disjoint in general but are allowed toshare edges and points.To set the stage for Theorem 2.3 and the proofs in Section 2.2, we introduce minors: An abstract graph G (cid:48) is a minor of G if it is obtained from G by a sequence of deleting and contracting edges and deleting isolatedvertices. Similarly, a spatial graph G (cid:48) is a minor of a spatial graph G if G (cid:48) is obtained from G by deletionand contraction of edges and deletion of isolated vertices. Contraction along an edge e of a spatial graph G means shrinking e to a point while keeping the edges which are attached to the endpoints of e attached.Contraction of an edge is only defined for edges which are not loops. Theorem 2.3 (Planarity criterion [12]) . A graph is planar if and only if it contains neither K nor K , as minors. Outline of the proof of Theorem 1.2:
The idea of the proof is to see that a simple 3-connected planar spatial graph G which is nontriviallyembedded in T contains a chiral subgraph which cannot be extended to an achiral spatial graph by addingvertices and edges on the torus without losing its planarity. Theorem 1.3 ensures the existence of nontriviallyknotted or nonsplit linked subgraphs of G . This allows a proof in two cases: Case 1 deals with the case inwhich G contains a nontrivial knot whereas Case 2 covers the case in which G contains a nonsplit link. Thesecond case has to further distinguish between the possibilities that either the nonsplit link is different fromthe Hopf link (Case 2a) or the Hopf link (Case 2b).In Case 1, the knot is chiral by Theorem 2.1. For G to be achiral, the mirror image of the knot must alsobe a subgraph of G by Remark 2.2. Therefore, we extend any nontrivial torus knot to a spatial graph K sothat the criterion of Remark 2.2 is satisfied. We show in Lemma 2.4 that K is a minor of every K which isconstructed in such a way. It follows from Theorem 2.3 that K is nonplanar. This proves Theorem 1.2 if G contains a nontrivial knot.In Case 2a, the same argument as for the knotted case proves the theorem. The only difference is thathere K , (instead of K ) is a minor of the extension K as shown in Lemma 2.5.In the remaining Case 2b, simpleness and 3-connectivity ensure the existence of the Hopf ladder withthree rungs (Fig. 1, right) as shown by [5]. The Hopf ladder with three rungs is chiral as shown in Proposi-tion 1.1. Here Lemma 2.6 ensures that K , is a minor of any achiral extension K of the Hopf ladder withthree rungs. he idea of the argument is similar to the one given by Castle, Evans and Hyde [5] who showed thestatement of Theorem 1.2 in the case that the spatial graph contains a nontrivial knot. But while the argu-ment in [5] uses the Euler characteristic, we detect one of Kuratowski’s nonplanar minors K , and K in theachiral extensions. There is a gap in the proof in [5] for the case that the spatial graph contains a nonsplitlink: The argument given there depends on the mistaken presumption that Simon [6] proved the chiralityof Hopf ladders with at least three rungs. Unfortunately, Simon’s proof assumes that rungs go to rungs andsides go to sides, which is not given in general. We fill the gap by showing the chirality of the Hopf ladderwith at least three rungs in Proposition 1.1 without making Simon’s extra assumptions. Proposition 1.1
The Hopf ladder H n with n rungs is achiral if ≤ n ≤ , chiral if ≤ n . Proof.
The Hopf link H is achiral as unoriented link but chiral if oriented, which can be easily confirmedby calculating the linking number. One or two rungs do not determine an orientation on the link. Sincethere exists a symmetric representation of the Hopf ladder with one or two rungs (Fig. 1, left), these Hopfladders are achiral. We prove the chirality of the Hopf ladder H n with n ≥ h of R whichmaps the spatial graph onto itself. Assume there exists an orientation-reversing homeomorphism h of R with h ( H n ) = H n , n ≥
3. Every homeomorphism h maps nontrivially linked subgraphs onto nontriviallylinked subgraphs. Since the Hopf link H which consists of the sides of H n is the only nontrivially linkedsubgraph of H n , it follows that H is mapped onto itself, i.e., h ( H ) = H . Since h reverses the orientation,we have h ( H ) = H (cid:63) , where H (cid:63) is the mirror image of H . The Hopf ladder H is chiral since more thantwo rungs e , . . . , e n with endpoints v , v (cid:48) , . . . , v n , v (cid:48) n determine orientations on the sides of the Hopf ladder(Fig. 1, right: v , . . . , v n = , , v (cid:48) , . . . , v (cid:48) n = , ,
6) by the order of the endpoints of the rungs v , . . . , v n respectively v (cid:48) , . . . , v (cid:48) n , therefore implying that H is different from H (cid:63) . This contradicts the existence of h with H = h ( H ) = H (cid:63) and it follows that the Hopf ladder H n , with n ≥
3, rungs is chiral. (cid:3)
123 456 123 456132 456F
IGURE
1. Left: The Hopf ladder with less than three rungs is achiral. It is neither simplenor 3-connected. Right: The Hopf ladder with three or more rungs is chiral by Proposi-tion 1.1.2.2.
Proof of Theorem 1.2.
The following three lemmas will be used in the proof of Theorem 1.2 to showthat achiral extensions of knots, chiral links and H on the torus fail to be planar. Lemma 2.4. K is a minor of every spatial graph on the torus which contains both knots T ( p , q ) and T ( r , − s ) for somerelatively prime integers p ≥ , q ≥ and relatively prime integers r ≥ , s ≥ .Proof. Let T ( a , b ) ⊗ T ( c , d ) be the spatial graph which is constructed by embedding T ( a , b ) and T ( c , d )with minimal number of intersections in the torus and by adding vertices at the intersection points. (Forexample, T (2 , ⊗ T (2 , −
5) is drawn on the left of Fig. 2 and T (2 , ⊗ T (1 , −
1) is drawn on the right.) Viewthe torus as the rectangle [0 , × [0 ,
1] with opposite sides identified. A meridian is given as { x } × [0 ,
1] anda longitude as [0 , × { y } . Without loss of generality, arrange T ( p , q ) and T ( r , − s ) on the rectangle in sucha way that no vertex lies on the boundary of the rectangle and that T ( r , − s ) runs through its corner point.Then there exists a path π in T ( p , q ) ⊗ T ( r , − s ) from the corner point (0 ,
1) to the corner point (1 ,
0) ofthe rectangle which does not intersect the boundary of the rectangle in any other points and which respectsthe orientations of T ( p , q ) and T ( r , − s ) (fat zig-zag in Fig. 2, left). The existence of π can be seen as follows compare Fig. 2 for notation): If r = s =
1, set π = T ( r , − s ). Otherwise, there are exactly two segments of T ( r , − s ) in the interior of the rectangle which are connected to the corner point. Denote by S the segmentwith endpoints (0 ,
1) and s and denote by S n the segment with endpoints s n and (1 , s lies on ]0 , ×{ } and s n on ]0 , ×{ } . Consider the region B of the rectangle which is bounded by the twosegments S and S n , [(0 , , s n ] and [ s , (1 , T ( r , − s ) ∩ ˚ B is eitherempty or the union of segments S i , 1 < i < n running parallel to S and S n , each having one endpoint in](0 , , s n [ and the other in ] s , (1 , B is a union of bands b i which are bounded by consecutivesegments S i and S i + , 1 ≤ i ≤ n −
1, and intervals in [(0 , , s n ] and [ s , (1 , S i and S i + . Since T ( p , q ) is a nontrivial torus knot, T ( p , q ) ∩ b i (cid:44) ∅ for all 1 ≤ i ≤ n −
1. Since T ( p , q )follows both the longitude and the meridian of the torus with positive orientation and since T ( r , − s ) followsthe longitude with positive and the meridian with negative orientation, T ( p , q ) ∩ b i does not run parallelto S i or S i + , and we conclude that there exists a connected component of T ( p , q ) ∩ b i which connects S i and S i + . This is true for all b i , ≤ i ≤ n −
1, hence a path from S to S n along T ( p , q ) ⊗ T ( r , − s ) inside]0 , × ]0 ,
1[ is found. Note that this path does not intersect the boundary of the rectangle since no vertex of T ( p , q ) ⊗ T ( r , − s ) lies on the boundary of the rectangle by assumption. Extending the path along S towards(0 ,
1) and along S n towards (1 , π . (Fig. 2, left). K ,
1) (1 , s s n S S n S S b b π b F IGURE
2. Illustration for Lemma 2.4. Left: T ( p , q ) ⊗ T ( r , − s ) with p = , q = , r = , s =
5. Here T ( p , q ) is drawn in light green (full), T ( r , − s ) in red (dashed), T (1 , −
1) inblue (full diagonal). A possible path π is drawn in dark blue (very fat). Second: Showingthe existence of the path π . B is the shaded region. Third: The graph K = S (2 , K represented as T (2 , ⊗ T (1 , −
1) is a minor of a spatial graph in the toruswhich contains both knots T ( p , q ) and T ( r , − s ). Blue (full): T (2 , T (1 , − T ( p , q ) ⊗ T ( r , − s ) as follows: Delete the edges of T ( r , − s ) which are not partof π , and contract the edges of T ( p , q ) ∩ π . This gives the spatial graph T ( p , q ) ⊗ T (1 , − S ( p , q ) (compare Fig. 3). S ( p , q ) can be described as a cycle C on which p + q vertices { v , . . . v p + q − } are placed, together with additional p + q edges c j which are chords connectingthe vertices v j and v j + p mod( p + q ), 0 ≤ j ≤ ( p + q − S ( p , q ) = S ( q , p ).It is left to show that K is a minor of S ( p , q ), which itself is a minor of T ( p , q ) ⊗ T ( r , − s ) (Fig. 3).Note that T ( q , p ) ⊗ T (1 , −
1) leads to the same graph S ( p , q ), allowing us to restrict the argument to graphs T ( p , q ) ⊗ T (1 , −
1) with p < q . Also, recall that p , q are relatively prime integers and p ≥
2. The argumentworks inductively. We present the argument in its entirety and add the details of the two single steps ‘takinga minor’ and ‘inversion plus renaming’ at the end of the proof.Set p = p , q = q . If p = , q =
3, we are done since S (2 , = K . For p i >
2, we constructa proper minor (i.e., a minor which is not the graph itself) of S ( p i , q i ) which is of the form S ( p i , M , q i , M )with p i , M = , q i , M = (cid:106) p i + q i p i (cid:107) −
1. By interchanging the roles of the cycle C i , M from S ( p i , M , q i , M ) andits chords, another representation S ( p (cid:48) i , M , q (cid:48) i , M ) of S ( p i , M , q i , M ) is obtained. This is possible because thechords are not edges of the cycle C i , M but form a closed cycle themselves by construction: S ( p , q ) is theCayley graph of the cyclic group ( Z / ( p + q ) , + ) with p + q elements and generator set { , p } . Since p is agenerator of Z / ( p + q ) and p , q are relatively prime integers, there exists a unique integer 0 ≤ k < p + q for each 0 ≤ j ≤ p + q − kp mod( p + q ) = j . Relabel the vertices of the graph by v j (cid:55)→ v k .In particular, there exists p (cid:48) with p (cid:48) p mod( p + q ) =
1, and if we set ( q + p ) − p (cid:48) = q (cid:48) , the relabeledgraph is of the form S ( p (cid:48) , p (cid:48) + (( q + p ) − p (cid:48) )) = S ( p (cid:48) , p (cid:48) + q (cid:48) ). S ( p (cid:48) i , M , p (cid:48) i , M ) and S ( p i , M , q i , M ) satisfy the elations p i , M p (cid:48) i , M mod( p i , M + q i , M ) = q i , M q (cid:48) i , M mod( p i , M + q i , M ) =
1. We show that we have either p (cid:48) i , M (cid:44) (cid:44) q (cid:48) i , M or we have p i , M = q i , M =
3. In the second case we are done, whereas in the firstcase we have completed the generic induction step: We now consider S ( p i + , q i + ), which is obtained from S ( q (cid:48) i , M , p (cid:48) i , M ) = S ( p (cid:48) i , M , q (cid:48) i , M ) by setting p i + : = q (cid:48) i , M , q i + : = p (cid:48) i , M . At each step, we reduce the numbers ofvertices and edges in the graph, namely p i + q i and 2( p i + q i ), respectively, when a minor is taken, i.e., p i − , M + q i − , M = p i + q i > p i , M + q i , M = p i + + q i + . In particular, since p i , M =
2, it follows that q i , M > q i + , M and that for all i the q i , M are odd numbers. We stop as soon as we either reach p i = , q i = S ( p i , q i ) = S ( p i , M , q i , M ), or we have p i , M = q i , M = S ( p i , M , q i , M ) = S ( p i + , q i + ). In any case, we reach S (2 ,
3) after finitely many steps: Since S (2 , = K , we are done. The steps of the iteration can be pictured as follows: S ( p i , q i ) minor −−−−→ S ( p i , M , q i , M ) invert −−−−→ S ( p (cid:48) i , M , q (cid:48) i , M ) = S ( q (cid:48) i , M , p (cid:48) i , M ) rename −−−−−−→ S ( p i + , q i + ) . ‘Taking a minor’: obtaining S ( p i , M , q i , M ) from S ( p i , q i ) (Fig. 3): Take the subgraph of S ( p i , q i ) whichconsists of C i and the first 2 (cid:106) p i + q i p i (cid:107) + p i =
2, wemust do exactly two full turns along chords. Only for p i =
2, the endpoints of the chords from the secondround lie exactly in the middle of the points to which the chords from the first round are attached. If theendpoints of the chords from the second round are closer to the endpoints than to the starting points of thechords from the first round, we have to stop traveling along chords just after completing the second roundin order to ensure that every chosen chord is intersected by exactly two other chords. If the endpoints of thechords from the second round are closer to the starting points than to the endpoints of the chords from thefirst round, we have to stop traveling along chords just before finishing the second round. We can in factdetermine in which case we are: (cid:106) p i + q i p i (cid:107) is the number of chords before finishing the first round. Therefore, (cid:106) p i + q i p i (cid:107) p i is the index of the endpoint of the last chord from the first round. If this vertex is closer to v thanto v q i + , we are in the first case, i.e., ( p i + q i ) − (cid:106) p i + q i p i (cid:107) p i < (cid:106) p i (cid:107) . Similarly, we are in the second case if( p i + q i ) − (cid:106) p i + q i p i (cid:107) p i > (cid:106) p i (cid:107) . If ( p i + q i ) − (cid:106) p i + q i p i (cid:107) p i = (cid:106) p i (cid:107) , we have p i =
2. Hence, in both cases, we havetraveled through the first 2 (cid:106) p i + q i p i (cid:107) + c , . . . c (cid:22) pi + qipi (cid:23) .To obtain S ( p i , M , q i , M ) from S ( p i , q i ) first delete the chords c (cid:22) pi + qipi (cid:23) + , . . . , c p i + q i − . Then contract iterativelyall edges of the cycle C i which have an endpoint that is not an endpoint of a remaining chord. Contractthe edge in C i which connects v and the endpoint of c (cid:22) pi + qipi (cid:23) . The resulting graph has 2 (cid:106) p i + q i p i (cid:107) + (cid:106) p i + q i p i (cid:107) +
1) edges by construction. Again, 2 (cid:106) p i + q i p i (cid:107) + C i , M , whereas theother edges are chords c j connecting the vertices v j and v j + mod(2 + q i , M ) for 0 ≤ j ≤ (2 + q i , M − q i , M = (cid:106) p i + q i p i (cid:107) −
1. Therefore, the graph is of the form S ( p i , M , q i , M ) and p i , M = S ( p i + , q i + ) from S ( p i , M , q i , M ) (Fig. 3): Since the chords are notedges of C i , M but form a closed cycle, we can change the roles of the chords and the cycle while leavingthe unlabeled graph unchanged. This leads to a representation of the graph in terms of p (cid:48) i , M and q (cid:48) i , M , whichare given by p i , M p (cid:48) i , M mod( p i , M + q i , M ) = q i , M q (cid:48) i , M mod( q i , M + q i , M ) =
1, as explained above. Note that p (cid:48) i , M > q (cid:48) i , M since p i , M < q i , M and p i , M + q i , M = p (cid:48) i , M + q (cid:48) i , M . To continue the induction we need to showthat p (cid:48) i , M (cid:44) q (cid:48) i , M (cid:44) p i , M = q i , M =
3, in which case we have already obtainedthe final graph S (2 , p (cid:48) i , M =
2, we have p i , M p (cid:48) i , M mod( p i , M + q i , M ) = · + q i , M ) = q i , M =
1, which contradicts the assumption that q i , M >
1. On the other hand, if q (cid:48) i , M =
2, wehave q i , M q (cid:48) i , M mod( p i , M + q i , M ) =
1, which is equivalent to q i , M q (cid:48) i , M = n ( p i , M + q i , M ) + n .Since q (cid:48) i , M = q i , M = n ( p i , M + q i , M ) +
1, and since p i , M = q i , M is apositive odd integer, this is equivalent to q i , M = n + n − with n =
1. Consequently, q i , M = S ( p i , M , q i , M ) = S (2 , K is a minor of S ( p , q ), which in turn is a minor of T ( p , q ) ⊗ T ( r , − s ). The obtained embeddingof K in the torus is T (2 , ⊗ T (1 , − (cid:3) (2 , = K S ( p , q ) S ( p , q ) S ( p M , q M ) v p v v p + q − v (cid:98) p + qp (cid:99) p v − p = q + v p S (2 , v v v v v v v v v v v v S (3 , v v v v v v v S (4 , v v v v v v v = S (2 , = S (3 , = S (4 , v v v S ( p − M , q − M ) = S (5 , IGURE
3. Illustration for Lemma 2.4. Top: the graph S ( p , q ) and examples for the cases( p + q ) − (cid:98) p + qp (cid:99) p = (cid:98) p (cid:99) , ( p + q ) − (cid:98) p + qp (cid:99) p > (cid:98) p (cid:99) , ( p + q ) − (cid:98) p + qp (cid:99) p < (cid:98) p (cid:99) . Bottom: Theprocedure of constructing the minor K from S (3 , Lemma 2.5. K , is a minor of every spatial graph on the torus which contains both links T ( kp , kq ) and T ( kp , − kq ) forsome relatively prime integers p and q and some k ≥ .Proof. Let p and q be relatively prime integers and let k ≥
2. Then T ( kp , − kq ) contains two differentconnected components of the form T ( p , − q ), which we denote by c and c , respectively. The components c and c run parallel on the torus. Likewise, T ( kp , kq ) contains two different connected componentsdenoted by c (cid:48) and c (cid:48) of the form T ( p , q ), which run parallel to each other (see Fig. 4 for notations). Since T ( p , q ) follows the longitude and the meridian of the torus with the same orientation and T ( p , − q ) followsthe longitude and the meridian with opposite orientation, c (cid:48) intersects c in at least two points. Take onesegment ( p , p ) of c (cid:48) which intersects c only in its endpoints denoted by p and p . If one follows c from p , the next intersection point denoted by p is with c (cid:48) . Let ( p , p ) be the segment of c . Then there startsa segment ( p , p ) of c (cid:48) which runs parallel to ( p , p ) and intersects c only in its endpoints p and p . Let( p , p ) be the segment of c which runs from p to p . This determines a disc in the torus which is boundedby the edges ( p , p ), ( p , p ), ( p , p ), and ( p , p ). Since c runs parallel to c , it cuts the boundary of thedisc in exactly two points p on ( p , p ) and p on ( p , p ). Following c , one passes through the points p , p , p , p in this order. This determines the subgraph with the six vertices p , . . . , p and the edges( p , p ), ( p , p ), ( p , p ), ( p , p ), ( p , p ), ( p , p ), ( p , p ), ( p , p ) and ( p , p ), which is precisely K , .Being a subgraph, K , is in particular a minor of every spatial graph on the torus which contains both links T ( kp , kq ) and T ( kp , − kq ). (cid:3)
31 2 45 6 c c (cid:48) c (cid:48) c
31 2 45 6 31 2 45 6F
IGURE
4. Illustration of Lemma 2.5. Left: the spatial graph T ( kp , kq ) ⊗ T ( kp , − kq ),with p = , q = , k =
2. Middle and right: K , is a minor of T ( kp , kq ) ⊗ T ( kp , − kq ). Lemma 2.6. K , is a minor of every achiral spatial graph on the torus which contains the Hopf ladder H . roof. Recall that a spatial graph is achiral if and only if there exists an orientation-reversing homeomor-phism of R which maps the spatial graph to itself. Such a homeomorphism induces a graph isomorphismon the graph. Graph isomorphisms map cycles to cycles, and graph isomorphisms which are induced byorientation-reversing homeomorphisms invert the sign of the linking number of links and their images.Consequently, every achiral graph which contains an oriented link also contains a second link which is itsmirror image. The sides of the Hopf ladder H form an oriented Hopf link, without loss of generality real-ized by T (2 , H as a subgraph from inheriting the chirality of the oriented Hopf link: Firstly, itcan contain a second oriented link T (2 , −
2) which is the mirror image of T (2 , T (2 , −
2) hasto be added to H since T (2 ,
2) is its only nontrivially linked subgraph. Since such an extension contains T (2 , ⊗ T (2 , − K , is a minor of this extension. Or, secondly, theorientation of the Hopf link can be eliminated by adding a further rung to the Hopf ladder. In that case, wefind that the resulting abstract graph is nonplanar, which is independent of the choice of embedding (Fig. 5).An extra edge with vertices 0 and Z is added whose vertex 0 lies on an edge of one component of the Hopflink – without loss of generality on the edge with endpoints 1 and 2, and whose vertex Z lies in the interiorof one of the edges with endpoint 3 on the other component of the Hopf link – without loss of generality onthe edge with vertices B and C . The Hopf ladder H with the extra edge (0 , Z ) contains K , as a minor ascan be seen by deleting the edge between 1 and 3 and contracting the edges between 0 and 1 and between2 and 3 (Fig. 5). Although not every embedding of this extension is an achiral graph, every extension of H which does not determine an orientation of the Hopf link contains the Hopf ladder H with an extra edge asconstructed above. It follows that every extension of H to an achiral graph contains a minor K , . (cid:3) CB AZ , CB AZ A , C B , ZK , , IGURE
5. Extending the abstract graph of the Hopf ladder H by adding the vertices 0and Z and an edge between them as described in Lemma 2.6. The extra edge inhibits theinduction of orientations on the two black circles which correspond to the Hopf link. Remark 2.7.
It is possible to embed the Hopf ladder H with the extra edge (1 , Z ) (as constructed in theproof of Lemma 2.6) as an achiral graph since there is a symmetric representation with the unoriented Hopflink mapped onto itself. The resulting spatial graph still lies on the torus but is nonplanar since K , is aminor (Fig. 6). It is shown in Fig. 6 how one is led to adding an extra rung when constructing the symmetricrepresentation by extending H to an achiral graph which maps the Hopf link to its mirror image. We are now ready to prove Theorem 1.2.
Proof.
It follows from Theorem 1.3 that the spatial graph G contains a subgraph which is a nontrivial knotor a nonsplit link. Case 1:
The spatial graph G contains a nontrivial knot.A subgraph of a spatial graph on the torus which forms a knot is a torus knot by definition. A nontrivialtorus knot can be written as T ( p , q ), for some relatively prime integers p and q with | p | , | q | > T ( − p , q ) = T ( p , − q ). If G is achiral, G contains both T ( p , q ) and T ( p , − q ) as subgraphs by Remark 2.2. As shown inLemma 2.4, K is a minor of every spatial graph which is embedded in the torus and which contains bothknots T ( p , q ) and T ( r , − s ) for some relatively prime integers p ≥ , q ≥ r ≥ , s ≥
1. This includes the case of a spatial graph containing both T ( p , q ) and T ( p , − q ) as subgraphs.The statement of Theorem 1.2 follows from Lemma 2.4 since by Theorem 2.3 a graph is nonplanar if andonly if it contains neither K nor K , as minors. A C B Z A C B , ZAB C Z K , CBAZ F IGURE
6. Extension of the Hopf ladder H to an achiral graph which contains both H and its mirror image H (cid:63) by adding one additional edge. Case 2:
The spatial graph G contains a nonsplit link.A subgraph of a spatial graph on the torus which forms a nonsplit link is a torus link by definition. Anonsplit torus link with k components can be written as T ( kp , kq ) for some relatively prime integers p and q with k ≥ Case 2a:
The spatial graph G contains a nonsplit link different from the Hopf link.A nonsplit torus link which is not the Hopf link is chiral by Theorem 2.1, and its mirror image is T ( − kp , kq ) = T ( kp , − kq ). If G is achiral, G contains both T ( kp , kq ) and T ( kp , − kq ) as subgraphs by Remark 2.2. Asshown in Lemma 2.5, K , is a minor of every graph G which is embedded in the torus and which containsboth links T ( kp , kq ) and T ( kp , − kq ), k ≥ Case 2b:
The spatial graph G contains a Hopf link.In this case 3-connectivity and simpleness imply that the Hopf ladder H with three rungs is a subgraphof G . By Proposition 1.1, H is chiral. If G is achiral, K , is a minor of G as shown in Lemma 2.6. Thestatement of Theorem 1.2 now follows from Lemma 2.6 and Theorem 2.3. (cid:3) Remark 2.8.
Simpleness and 3-connectivity in Theorem 1.2 are only needed for the case in which G con-tains no nontrivial knot and the only nontrivially linked subgraph forms a Hopf link. If the spatial graphcontains a nontrivial knot or a nonsplit link different from the Hopf link, these two assumptions can bedropped. Remark 2.9.
It is not possible to weaken the assumptions of Theorem 1.2. Counterexamples are given orcan be constructed as in [5] . A CKNOWLEDGMENTS
I thank Matt Rathbun, Stephen Hyde and the anonymous reviewer for helpful comments and discussions.I thank David Rottensteiner for proof-reading and helping me organize the proof of Lemma 2.4. The earlystage of this research was supported by the Roth studentship of Imperial College London mathematicsdepartment, the DAAD, and the Evangelisches Studienwerk.R
EFERENCES[1] S. Barthel, There exist no Minimally Knotted Planar Spatial Graphs on the Torus,
J. of Knot Theory Ramif. (2015) 1550035–1550042, arXiv:1411.7734.[2] S. Barthel and D. Buck, Toroidal embeddings of abstractly planar graphs are knotted or linked, J. Math. Chem. (2015) 1–19,arXiv:1505.05696.[3] K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topology (1987) 187–194.
4] K. Murasugi, Jones polynomials and classical conjectures in knot theory II,
Math. Proc. Camb. Phil. Soc. (1987) 317–318.[5] T. Castle, M. E. Evans and S. T. Hyde, All toroidal embeddings of polyhedral graphs in 3-space are chiral,
New J. Chem. ,(2009) 2107–2113.[6] J. Simon, A topological approach to the stereochemistry of nonrigid molecules in Graph Theory and Topology in Chemistry , eds.R. B. King, D. H. Rouvray, (Elsevier, Amsterdam, 1987), pp. 43–75.[7] J.-P. Sauvage and D. B. Amabilino, Templated Synthesis of Knots and Ravels in
Supramolecular Chemistry: From Molecules toNanomaterials Vol5 , eds. P. Gale and J. Steed (Wiley, 2012).[8] R. S. Forgan, J.-P. Sauvage and J. F. Stoddart, Chemical Topology: Complex Molecular Knots, Links, and Entanglements,
Chem. Rev. (2011) 5434–5464.[9] D. B. Amabilino and L. P´erez-Garc´ıa, Topology in molecules inspired, seen and represented
Chem. Soc. Rev. (2009) 1562–1571.[10] L. Carlucci, G. Ciani and D. M. Proserpio, Polycatenation, polythreading and polyknotting in coordination network chemistry, Coord. Chem. Rev. (2003) 247–289.[11] E. Flapan,
When Topology meets Chemistry ( Camb. Univ. Press, 2000).[12] K. Kuratowski, Sur le probl`eme des courbes gauches en topologie,
Fund. Math. (1930) 271–283.EPFL V ALAIS , R
UE DE L ’I NDUSTRIE
17, 1951 S
ION , S
WITZERLAND
E-mail address : [email protected]@epfl.ch