On the Wu invariants for immersions of a graph into the plane
aa r X i v : . [ m a t h . G T ] F e b ON THE WU INVARIANTS FOR IMMERSIONS OF A GRAPHINTO THE PLANE
RYO NIKKUNI
Abstract.
We give an explicit calculation of the Wu invariants for immersionsof a finite graph into the plane and classify all generic immersions of a graphinto the plane up to regular homotopy by the Wu invariant. This result is ageneralization of the fact that two plane curves are regularly homotopic if andonly if they have the same rotation number. Introduction
Throughout this paper we work in the piecewise linear category. In [7], [6], Wudefined an isotopy invariant of embeddings and immersions of polyhedra into theEuclidean space in terms of the cohomology of deleted product spaces. In case ofembeddings, this invariant classifies all embeddings of a graph into R up to spatialgraph-homology (Taniyama [4]). But as far as the author knows, little is knownabout an application of this invariant in case of immersions. Our purpose in thispaper is to give an explicit calculation of Wu’s invariant of immersions of a graphinto R and apply it to a geometric classification.Let G be a finite, connected and simple graph which has at least one edge. Wedenote the set of all vertices (resp. edges) of G by V ( G ) (resp. E ( G )). If theterminal vertices of an edge e of G are u and v , then we denote e = ( u, v ) = ( v, u ).We denote the number of edges incident to a vertex v by deg v . Note that G has astructure of a finite 1-dimensional simplicial complex. We regard G as a topologicalspace by considering its geometric realization, namely G is a compact and connected1-dimensional polyhedron. In this situation, each of the vertices and the edges of G can be regarded as a subset of G . We call a continuous map f : G → R a plane immersion of G if there exists an open covering { U ν } of G such that f | U ν isan embedding for any ν . A plane immersion f of G is said to be generic if all ofits multipoints are transversal double points away from vertices. We say that twoplane immersions f and g of G are regularly homotopic if there exists a homotopy F : G × [0 , → R from f to g and an open covering { U ν } of G such that f t | U ν isan embedding for any ν and for any t ∈ [0 , f t is a continuous map from G to R defined by f t ( x ) = F ( x, t ) for any x ∈ G . Note that regular homotopydefines an equivalence relation on plane immersions of a graph. We give the precise definition of the Wu invariant R ( f ) of a plane immersion f of a graph in the next section and also give an explicit calculation of R ( f ) insection 3. It can be calculated as a first cohomology class of a subspace of the Mathematics Subject Classification.
Primary 57Q35; Secondary 57M15.
Key words and phrases.
Immersion, Graph, Wu invariant. This equivalence relation was introduced in [6] by the name of local isotopy . symmetric deleted product of the graph, which is called the symmetric tube of thegraph. Moreover we have the following classification theorem. Theorem 1.1.
Let f and g be two generic plane immersions of a graph G . Thenthe following are equivalent: ( 1 ) f and g are regularly homotopic. ( 2 ) f and g are transformed into each other by the local moves as illustrated inFig. 1.1 (1), (2), (3) and ambient isotopies. ( 3 ) R ( f ) = R ( g ) . (1)(2)(3) Figure 1.1.
Let K m be the complete graph on m vertices for a positive integer m , namely V ( K m ) = { v , v , . . . , v m } and E ( K m ) = { ( v i , v j ) (1 ≤ i < j ≤ m ) } . A planeimmersion f of K is called a plane curve . By Theorem 1.1, we have the followingcorollary. Corollary 1.2.
Let f and g be two generic plane curves. Then the following areequivalent: ( 1 ) f and g are regularly homotopic. ( 2 ) f and g are transformed into each other by the local moves as illustrated inFig. 1.1 (1), (2) and ambient isotopies. ( 3 ) R ( f ) = R ( g ) . We prove Theorem 1.1 in section 4. As we will see in Example 3.11, R ( f ) of aplane curve f coincides with the rotation number [5] of f . (In this paper we con-sider the orientation on R with positive rotation numbers in the counterclockwisedirection.) Thus Corollary 1.2 coincides with the regular homotopy classificationof plane curves by Whitney-Graustein’s theorem [5] and Kauffman’s combinatorialinterpretation [2]. We remark here that recently Permyakov gives a simple combi-natorial interpretation of R ( f ) and shows a theorem which corresponds to Theorem1.1 [3]. N THE WU INVARIANTS FOR IMMERSIONS OF A GRAPH INTO THE PLANE 3 Wu invariant
In this section we give the definition of the Wu invariant of a plane immersion ofa graph G . We refer the reader to [6] for the general case. Let X be a topologicalspace. For the embedding ˜ d : X → X × X defined by ˜ d ( x ) = ( x, x ), we call e X ∗ = ( X × X ) \ ˜ d ( X ) the deleted product of X . A map σ ( x, y ) = ( y, x ) gives a free Z -action on e X ∗ . We call X ∗ = e X ∗ / Z the symmetric deleted product of X . Wedenote the image of ˜ d ( X ) by the natural projection from X × X to ( X × X ) / Z by d ( X ). Let U be a neighborhood of ˜ d ( X ) in X × X . Then e U ∗ = U \ ˜ d ( X ) iscalled a deleted neighborhood of ˜ d ( X ) in e X ∗ . A deleted neighborhood e U ∗ is saidto be σ -invariant if σ ( U ) = U . Then we call U ∗ = e U ∗ / Z a symmetric deletedneighborhood of d ( X ) in X ∗ .For a graph G , let { U ∗ λ } be the set of all symmetric deleted neighborhoods of d ( G ) in G ∗ . Then { U ∗ λ , ≺} forms an oriented set by U ∗ λ ≺ U ∗ µ if U ∗ λ ⊃ U ∗ µ . Forthis oriented set, { H ( U ∗ λ ; Z ) , i µλ ∗ } forms an inductive system of modules, where H ( · ; Z ) denotes the integral first cohomology group and i µλ ∗ : H ( U ∗ λ ; Z ) → H ( U ∗ µ ; Z )is a homomorphism induced by the inclusion. Then we denote the inductive limitlim −→ H ( U ∗ λ ; Z ) by R ( G ). We note that we have the following natural homomorphism i ∗ λ : H ( U ∗ λ ; Z ) −→ R ( G )(2.1)for any symmetric deleted neighborhood U ∗ λ of d ( G ) in G ∗ .Let f : G → R be a plane immersion. Namely there exists an open covering U = { U ν } of G such that f | U ν is an embedding for any ν . Then the set f W U = { ( x , x ) ∈ e G ∗ | there exists a U ν such that x , x ∈ U ν } forms a σ -invariant deleted neighborhood of ˜ d ( G ) in e G ∗ and a continuous map¯ f : W U → ( R ) ∗ is defined by ¯ f [ x , x ] = [ f ( x ) , f ( x )]. On the other hand, it is wellknown that a continuous map r : ( e R ) ∗ → S defined by r ( y , y ) = ( y − y ) / || y − y || is a σ -equivariant strong deformation retract and r : ( R ) ∗ → S / Z ≈ S isalso a strong deformation retract, where S / Z denotes the quotient space of S byidentifying the antipodal points. Let Σ be a generator of H ( S ; Z ) ∼ = Z . Then theimage of Σ by the composition H ( S ; Z ) r ∗ ∼ = −→ H ( R ∗ ; Z ) ¯ f ∗ −→ H ( W U ; Z ) i ∗U −→ R ( G )is denoted by R ( f ), where i ∗U is the natural homomorphim of (2.1) for W U . Wecall R ( f ) a Wu invariant of f . We remark here that the definition above isindependent of the choice of U . Proposition 2.1. ([6]) R ( f ) is a regular homotopy invariant.Proof. Let f and g be two regularly homotopic plane immersions of G . Namelythere exists a homotopy F : G × [0 , → R from f to g and an open covering { U ν } of G such that f t | U ν is an embedding for any ν and for any t ∈ [0 , f t ( x ) = F ( x, t ) for x ∈ G . Then we can define a homotopy F U : W U × [0 , −→ ( R ) ∗ from¯ f to ¯ g by F U ([ x , x ] , t ) = [ f t ( x ) , f t ( x )]. Thus we have that R ( f ) = i ∗U ¯ f ∗ ¯ r ∗ (Σ) = i ∗U ¯ g ∗ ¯ r ∗ (Σ) = R ( g ). This completes the proof. (cid:3) This invariant was introduced in [6] by the name of local isotopy class and denoted by Λ f ( G ). RYO NIKKUNI Symmetric tube of a graph
A precise method to calculate R ( G ) is provided in [6]. Let X and Y be twotopological spaces and M = X ∪ ( X × Y × [0 , ∪ Y the disjoint union. Let usconsider a quotient space by identifying ( x, y, ∈ X × Y × [0 ,
1] with x ∈ X and( x, y, ∈ X × Y × [0 ,
1] with y ∈ Y . We call the quotient space a join of X and Y and denote it by X ◦ Y . We set [ X, Y ] (0) = { [ x, y, / ∈ X ◦ Y | x ∈ X, y ∈ Y } . Forexample, the join v ◦ e of a vertex v and an edge e is homeomorphic to a 2-simplex,and [ v, e ] (0) is homeomorphic to a 1-simplex. The following is a special case of whatis called a canonical cellular decomposition of the product space of X [6], [1]. Proposition 3.1.
Let G be a graph. Then G × G is decomposed into the followingcells: ( 1 ) ˜ d ( s ) for s ∈ V ( G ) or E ( G ) . ( 2 ) s × s for s i ∈ V ( G ) or E ( G ) ( i = 1 , and s ∩ s = ∅ . ( 3 ) ˜ d ( s ) ◦ ( s × s ) for s, s , s ∈ V ( G ) or E ( G ) , s ∩ s = ∅ and s ∪ s i iscontained in a vertex or an edge of G ( i = 1 , . In particular, the cellular complex which consists of all cells [ s, s × s ] (0) forsimplices s , s and s of Proposition 3.1 (3) is called a tube of G and is denotedby e G (0) . Clearly e G (0) is σ -invariant in e G ∗ . We call e G (0) / Z a symmetric tube of G and denote it by G (0) . We denote [ s, s × s ] (0) / Z by [ s, s ∗ s ], and we note that[ s, s ∗ s ] = [ s, s ∗ s ]. Example 3.2.
Let K be the complete graph on three vertices as illustrated inFig. 3.1. The figure on the right side in Fig. 3.1 illustrates the canonical cellulardecomposition of K × K in the sense of Proposition 3.1 as an expanded diagramof the 2-dimensional torus. The dotted thick parts and black thick parts representthe cells of Proposition 3.1 (1) and (2), respectively. The gray thick parts represent e K (0)3 . Thus we can see that K (0)3 is homeomorphic to the circle.Let P ( e G ∗ ) be the cellular complex which consists of all cells s × s for simplices s , s of Proposition 3.1 (2). Clearly P ( e G ∗ ) is also σ -invariant in e G ∗ . We denote P ( e G ∗ ) / Z by P ( G ∗ ). We note that G ∗ \ P ( G ∗ ) is a symmetric deleted neighborhoodof d ( G ) in G ∗ . It is known that i ∗ G ∗ \ P ( G ∗ ) : H ( G ∗ \ P ( G ∗ ); Z ) ∼ = −→ R ( G ) , where i ∗ G ∗ \ P ( G ∗ ) is the natural homomorphism of (2.1) for G ∗ \ P ( G ∗ ), and thereexists a deformation retract j : G ∗ \ P ( G ∗ ) → G (0) [6]. Therefore we have thefollowing. Theorem 3.3. ([6]) i ∗ G (0) = i ∗ G ∗ \ P ( G ∗ ) j ∗ : H ( G (0) ; Z ) ∼ = −→ R ( G ) . Thus the calculation of H ( G (0) ; Z ) provides a precise method to calculate R ( G ).To calculate H ( G (0) ; Z ), we investigate the structure of G (0) directly. We set V ( G ) = { v , v , . . . , v m } and E ( G ) = { e , e , . . . , e n } . We choose a fixed orientationon each edge of G . We put Z sst = Z sts = [ v s , v s ∗ v t ] for ( v s , v t ) ∈ E ( G ), W ust = W uts = [ v u , v s ∗ v t ] for ( v u , v s ) , ( v u , v t ) ∈ E ( G ), v s = v t , X ist = X its = [ e i , v s ∗ v t ] for e i = ( v s , v t ) ∈ E ( G ), and Y sti = [ v s , v t ∗ e i ] for ( v s , v t ) , e i ∈ E ( G ), ( v s , v t ) = e i and v s ⊂ e i . N THE WU INVARIANTS FOR IMMERSIONS OF A GRAPH INTO THE PLANE 5 v v v e e e K v v v v v v v v e e e e e e K K Figure 3.1.
Note that Z sst and W ust are 0-dimensional simplices of G (0) , and X ist and Y sti are1-dimensional simplices of G (0) . An orientation of X ist is induced by e i , and anorientation of Y sti is induced by e i . We can consider Z sst and W ust as 0-chains in C ( G (0) ; Z ) and X ist and Y sti as 1-chains in C ( G (0) ; Z ). The dual cochain of Z sst , W ust , X ist and Y sti are denoted by Z sts , W stu , X sti and Y tis , respectively. It is notdifficult to see the following. Proposition 3.4.
For a graph G , a cell of symmetric tube G (0) is one of Z sst , W ust , X ist and Y sti as above. Therefore G (0) is also a graph. For example, let S n be a graph as illustrated in Fig. 3.2. By enumerating verticesand edges of S (0) n and observing the connection between them directly, we have thefollowing. Lemma 3.5.
The symmetric tube S (0) n of S n is a graph as follows: ( 1 ) V ( S (0) n ) = { Z in +1 ,i and Z n +1 n +1 ,i ( i = 1 , , . . . , n ) , W n +1 jk (1 ≤ j < k ≤ n ) } . ( 2 ) E ( S (0) n ) = { X in +1 ,i ( i = 1 , , . . . , n ) , Y n +1 jk and Y n +1 kj (1 ≤ j < k ≤ n ) } . ( 3 ) X in +1 ,i = ( Z n +1 n +1 ,i , Z in +1 ,i ) ( i = 1 , , . . . , n ) , Y n +1 jk = ( Z n +1 n +1 ,j , W n +1 jk ) and Y n +1 kj = ( Z n +1 n +1 ,k , W n +1 jk ) (1 ≤ j < k ≤ n ) . ( 4 ) deg Z in +1 ,i = 1 and deg Z n +1 n +1 ,i = n ( i = 1 , , . . . , n ) , deg W n +1 jk = 2 (1 ≤ j < k ≤ n ) . Example 3.6.
Fig. 3.3 illustrates the symmetric tube S (0) n of S n for n = 1 , H of S (0) n defined by V ( H ) = { Z n +1 n +1 ,i ( i = 1 , , . . . , n ) , W n +1 jk (1 ≤ j < k ≤ n ) } ,E ( H ) = { Y n +1 jk , Y n +1 kj (1 ≤ j < k ≤ n ) } RYO NIKKUNI v v v v v n e v n+1 e e e e n Figure 3.2. is homeomorphic to K n . Precisely speaking, H is isomorphic to the graph which isobtained from K n by subdividing each edge of K n once.By Lemma 3.5 and Example 3.6, we can see that the symmetric tube G (0) of agraph G is obtained from G by “substituting” K d for each vertex v s of G as follows,where d = deg v s . Also see Examples 3.11 and 3.13. Lemma 3.7.
Let G be a graph. For a vertex v s of G , let v s , v s , . . . , v s d be allvertices connected to v s so that e i l = ( v s , v s l ) ( l = 1 , , . . . , d, ≤ i j < i k ≤ d for j < k ) . Then G (0) is obtained from G by replacing each v s with the graph H s ,which is homeomorphic to K d defined by V ( H s ) = { Z sss l ( i = 1 , , . . . , d ) , W ss j s k (1 ≤ j < k ≤ d ) } ,E ( H s ) = { Y ss j i k = ( Z sss j , W ss j s k ) , Y ss k i j = ( Z sss k , W ss j s k ) (1 ≤ j < k ≤ d ) } , and by replacing e i l with X i l ss l = ( Z s l ss l , Z sss l ) ( l = 1 , , . . . , d ) . By the universal coefficient theorem, it is sufficient to know H ( G (0) ; Z ) to cal-culate H ( G (0) ; Z ). So in the following we construct a spanning tree of G (0) . Firstwe define a subgraph T S (0) n of S (0) n as follows. We set V ( T S (0) n ) = V ( S (0) n ) and E ( T S (0) n ) = { X in +1 ,i ( i = 1 , , . . . , n ) , Y n +1 nj ( j = 1 , , . . . , n − ,Y n +1 tn ( t = 1 , , . . . , n − , Y n +1 kl (1 ≤ k < l ≤ n − } . We note that E ( S (0) n ) \ E ( T S (0) n ) = { Y n +1 lk (1 ≤ k < l ≤ n − } . Then we easily have the following.
Lemma 3.8.
A subgraph T S (0) n is a spanning tree of S (0) n . Now we construct a spanning tree of G (0) on the outcome of Lemma 3.8. For v s ∈ V ( G ), let st( v s ) be a subgraph of G consisting of v s and all edges incident to v s . Let v s , v s , . . . , v s d be all vertices connected to v s so that e i l = ( v s , v s l ) ( l =1 , , . . . , d, ≤ i j < i k ≤ d for j < k ), where d = deg v s . We construct a spanningtree T st( v s ) of st( v s ) in the same way as S (0) n . Namely, V ( T st( v s ) ) = V (st( v s )) and E ( T st( v s ) ) = { X i l ss l ( l = 1 , , . . . , d ) , Y ss d i j ( j = 1 , , . . . , d − ,Y ss j i d ( j = 1 , , . . . , d − , Y ss j i k (1 ≤ j < k ≤ d − } . N THE WU INVARIANTS FOR IMMERSIONS OF A GRAPH INTO THE PLANE 7 Z W Y S (0)3 v v v e e e S v Z Z Z Z Z W W X X X Y Y Y Y Y v e v Z X Z S (0)1 S v v e e v Z W Y Z Z Z X X Y S (0)2 S Figure 3.3.
Let T G be a spanning tree of G . We define a subgraph T G (0) of G (0) by V ( T G (0) ) = V ( G (0) ) and E ( T G (0) ) = { X jj j | e j = ( v j , v j ) ∈ E ( T G ) }∪ [ v s ∈ V ( G ) (cid:0) E ( T st( v s ) ) \ { X i l ss l ( l = 1 , , . . . , d ) } (cid:1) . Then we have the following.
Lemma 3.9.
A subgraph T G (0) is a spanning tree of G (0) . We note that E (st( v s )) \ E ( T st( v s ) ) = { Y ss j i k (1 ≤ j < k ≤ d − } RYO NIKKUNI for v s ∈ V ( G ) and this set is empty for d = 1 ,
2. Therefore we have that E ( G (0) ) \ E ( T G (0) ) = { X jj j | e j = ( v j , v j ) ∈ E ( G ) \ E ( T G ) }∪ [ vs ∈ V ( G )deg vs ≥ { Y ss j i k (1 ≤ j < k ≤ d − } . Thus we can determine a structure of H ( G (0) ; Z ) completely as follows. Theorem 3.10. ( 1 ) H ( G (0) ; Z ) ∼ = M ej ∈ E ( G ) \ E ( TG ) ej =( vj ,vj D X j j j E ⊕ M vs ∈ V ( G )deg vs ≥ M ≤ j 12 (deg v s − v s − n − m + 1 + 12 m X s =1 (cid:8) (deg v s ) − v s + 2 (cid:9) = n − m + 1 + 12 m X s =1 (deg v s ) − m X s =1 deg v s + m = n + 1 + 12 m X s =1 (deg v s ) − n = 1 − n + 12 m X s =1 (deg v s ) . This completes the proof. (cid:3) Example 3.11. Let K be the complete graph on three vertices as illustrated in theleft side of Fig. 3.4. As we saw in Example 3.2, the symmetric tube K (0)3 is a graphas illustrated in Fig. 3.4. For a spanning tree T K = e ∪ e of K , by Theorem 3.10we have that H ( K (0)3 ; Z ) = (cid:10) X (cid:11) ∼ = Z . We note that if [ x, y, / ∈ K (0)3 rotatesonce around the one in the direction induced by the orientation of X then thenon-ordered pair ( x, y ) rotates once around K ; see Fig. 3.5. Here the initial andterminal points of a vector in Fig. 3.5 correspond to x and y , respectively. Thisshows that R ( f ) of a plane curve f coincides with the rotation number of f . Example 3.12. For S and its symmetric tube S (0)3 as illustrated in Fig. 3.3, byTheorem 3.10 we have that H ( S (0)3 ; Z ) = (cid:10) Y (cid:11) ∼ = Z . We note that if [ x, y, / ∈ S (0)3 rotates once around the cycle represented by Y in the direction induced bythe orientation of the one then the non-ordered pair ( x, y ) rotates once around v ;see Fig. 3.6. Here the initial and terminal points of a vector in Fig. 3.6 correspondto x and y , respectively.Let f be a generic plane immersion of S . Then there exists a neighbourhood U of v such that f | U is an embedding. We can see that R ( f ) = 1 if f | U ( e ∩ U ), N THE WU INVARIANTS FOR IMMERSIONS OF A GRAPH INTO THE PLANE 9 Z W Z Z W Z Z W Z X Y X X Y Y Y Y Y K (0)3 v v v e e e K Figure 3.4. v v v e e e v v v e e e v v v e e e v v v e e e v v v e e e v v v e e e v v v e e e v v v e e e v v v e e e Figure 3.5. f | U ( e ∩ U ) and f | U ( e ∩ U ) are embedded in R as illustrated in Fig. 3.7 (1), and R ( f ) = − f | U ( e ∩ U ), f | U ( e ∩ U ) and f | U ( e ∩ U ) are embedded in R asillustrated in Fig. 3.7 (2). Example 3.13. Let K be the complete graph on four vertices and f , g and h threegeneric plane immersions of K as ilustrated in Fig. 3.8. Then the symmetric tubeof K is a graph as illustrated in Fig. 3.9. For a spanning tree T K = e ∪ e ∪ e 30 RYO NIKKUNI v v v e e e v v v v e e e v v v v e e e v v v v e e e v v v v e e e v v v v e e e v v v v e e e v Figure 3.6. v f (e U) U U f (e U) U U f (e U) U U f ( ) U v f (e U) U U f (e U) U U f (e U) U U f ( ) U (1) (2) Figure 3.7. of K , by Theorem 3.10 we have that H ( K (0)4 ; Z ) = (cid:10) X , X , X , Y , Y , Y , Y (cid:11) ∼ = Z ⊕ Z ⊕ · · · ⊕ Z | {z } . By calculating on the outcome of Examples 3.11 and 3.12, we have that R ( f ) = ( − , , − , , , , , R ( g ) = (1 , − , , , − , , − 1) and R ( h ) = (0 , , , , , − , . Proof of Theorem 1.1 First we show two lemmas which are needed to prove Theorem 1.1. N THE WU INVARIANTS FOR IMMERSIONS OF A GRAPH INTO THE PLANE 11 v v v v e e e e e e f(v ) f(v ) f(v ) f(v ) g(v ) g(v ) g(v ) g(v ) h(v ) h(v ) h(v ) h(v ) f g h Figure 3.8. Z Z W W W W W W W W W W W W Z Z Z Z Z Z Z Z Z Z X X X X X X Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Figure 3.9. K (0)4 Lemma 4.1. Each of the local moves as illustrated in Fig. 4.1 (4) and (5) arerepresented by a sequence of moves from the list as illustrated in Fig. 1.1 (1), (2),(3) and ambient isotopies.Proof. See Fig. 4.2 and 4.3. (cid:3) We remark here that the local move as illustrated in Fig. 4.1 (4) is none otherthan the Whitney trick. (4)(5) Figure 4.1. (2) (1)(1) Figure 4.2. (3) (1), (2)(1), (2) Figure 4.3. Lemma 4.2. Let G be a graph, H a connected subgraph of G and f and g twoplane immersions of G . If R ( f ) = R ( g ) , then R ( f | H ) = R ( g | H ) . N THE WU INVARIANTS FOR IMMERSIONS OF A GRAPH INTO THE PLANE 13 Proof. Let i : H → G be the inclusion. Since i is injective, the homomorphism i ∗ : H ( G (0) ; Z ) −→ H ( H (0) ; Z )is induced by i . Clearly i ∗ ( R ( f )) = R ( f | H ) and i ∗ ( R ( g )) = R ( g | H ). Therefore bythe assumption we have the desired conclusion. (cid:3) For a generic plane immersion f and a vertex v s of a graph G , a cyclic order ofthe edges of G incident to v s is determined by considering a neighbourhood U of v s so that f | U is an embedding. We call it a cyclic order of f ( v s ). Proof of Theorem 1.1. Since (1) ⇒ (3) is shown by Proposition 2.1 and (2) ⇒ (1) isclear, it is sufficient to show that (3) ⇒ (2). Assume that R ( f ) = R ( g ). In thefollowing we show that f and g are transformed into each other by the moves asillustrated in Fig. 1.1 (1), (2), (3), Fig. 4.1 (4), (5) and ambient isotopies. Then byLemma 4.1, we have the desired conclusion. Since R ( f ) = R ( g ), by Lemma 4.2 wehave that R ( f | st( v s ) ) = R ( g | st( v s ) ) for any vertex v s of G . Then by Example 3.12we have that the cyclic order of f ( v s ) is equal to the cyclic order of g ( v s ) for anyvertex v s of G . Let T G be a spanning tree of G . By using the moves as illustratedin Fig. 1.1 (1), (2), (3), Fig. 4.1 (5) and ambient isotopies in case of necessity,we can deform f (resp. g ) so that f | T G (resp. g | T G ) is an embedding. Since thecyclic order of f ( v s ) is equal to the cyclic order of g ( v s ) for any vertex v s of G , wemay assume that f | T G = g | T G . We set E ( G ) \ E ( T G ) = { e k , e k , . . . , e k β } , where β denotes the first Betti number of G . Let p k i be the unique path on T G whichconnects the terminal vertices of e k i . We denote a cycle e k i ∪ p k i by γ k i . Note thatthe double points of f | γ ki (resp. g | γ ki ) are only the double points of f | e ki (resp. g | e ki ). Then, by using the moves as illustrated in Fig. 1.1 (1), (2), (3), Fig. 4.1(4) and ambient isotopies in case of necessity, we can deform f | γ ki into the genericplane immersion of γ k i as illustrated in Fig. 4.4 (1) or (2) ( i = 1 , , . . . , β ). Then,by Lemma 4.2 we have that R ( f | γ ki ) = R ( g | γ ki ), namely f | γ ki and g | γ ki have thesame rotation number. Thus we may assume that f | γ ki = g | γ ki for i = 1 , , . . . , β .This implies that we can deform f and g identically by the moves as illustrated inFig. 1.1 (1), (2), (3), Fig. 4.1 (4), (5) and ambient isotopies. This completes theproof. (cid:3) f(e ) k i f(p ) k i f(e ) k i f(p ) k i (1) (2) Figure 4.4.Acknowledgment The author is most grateful to Professors Tohl Asoh and Tsutomu Yasui for theirinvaluable comments. References [1] S. T. Hu, Isotopy invariants of topological spaces, Proc. Roy. Soc. London. Ser. A (1960),331–366.[2] L. H. Kauffman, Formal knot theory , Mathematical Notes , Princeton University Press,Princeton, NJ, 1983.[3] D. A. Permyakov, Classification of immersions of graphs into a plane, Moscow Univ. Math.Bull. (2008), 208–210.[4] K. Taniyama, Homology classification of spatial embeddings of a graph, Topology Appl. (1995), 205–228.[5] H. Whitney, On regular closed curves on the plane, Compositio Math. (1937), 276–284.[6] W. T. Wu, A theory of Imbedding, Immersion, and Isotopy of Polytopes in a EuclideanSpace , Science Press, Peking, 1965.[7] W. T. Wu, On the isotopy of a complex in a Euclidean space. I, Sci. Sinica (1960), 21–46. Department of Mathematics, School of Arts and Sciences, Tokyo Woman’s ChristianUniversity, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan E-mail address ::