The warping degree and the unknotting number of a spatial graph Akio Kawauchi Osaka City University at East Asian School at Gyeongju January 15, 2009

Contents １. A spatial graph without degree one vertices and its diagram 2. A monotone diagram 3. Complexity 4. Warping degree and unknotting number 5. A spatial graph with degree one vertices

１. A spatial graph without degree one vertices and its diagram Let Γ = a (possibly disconnected) finite graph with (1) only vertices of degree ≧ 2 and (2) at most one component containing vertices of degrees ≧ 3.

Definition. A spatial graph of Γ is an embedding image G of Γ into R 3 such that ∃ an orientation-preserving homeomorphism h: R 3 → R 3 such that h(G ) is a polygonal graph in R 3.

Definition. A diagram D=D G of a spatial graph G is the image p a (G) under a plane projection p a : R 3 → P along a norm 1 vector a in R 3 such that the singularities are only double point singularities together with the upper-lower crossing information on every double point.

2. A monotone diagram In a spatial graph G, we ignore the degree 2 vertices. Fact: Every spatial graph G (ignored the degree 2 vertices) is obtained from a maximal tree T by adding some edges or loops α i (i=1,2,…,m). Note: (1) T ＝ φ when G is a link. (2) T = one vertex if G has just one vertex.

Let D be a diagram of a spatial graph G of Γ, and D T, D α i the subdiagrams of D corresponding to T and α i, respectively. Let c D (D T ) be the number of crossing points of D belonging to D T. Definition. D is a based diagram (on T ), written as (D;T) if c D (D T )=0.

⇒ Lemma. D is deformed into a based diagram by moves on D T. An edge diagram D α in a diagram D is monotone if:

Definition. A based diagram (D;T) is monotone if D α i is monotone and D α i is upper than D α j for i < j in an ordered sequence of the remaining edge diagrams D α i (i=1,2,…,m) of T.

Definition. The warping degree d(D;T) of a based diagram (D;T) is the least number of crossing changes needed to obtain a monotone diagram from D. 3. Complexity

Note. (1)(M. Ozawa, A. Shimizu). For the warping degree of an oriented edge diagram D α, (D α ) + (-D α )= c(D α ), d(D α ) = min{ (D α ), (-D α )}. For example, d( ) =1, for =1: =3: (2) (A. Shimizu). For an oriented knot diagram D with T=φ, (D)+ (-D) ≦ c(D)-1, where the equality holds if and only if D is an alternating diagram. →d→d →d→d →d→d →d→d →d→d →d→d →d→d →d→d →d→d

Definition. The complexity of a based diagram (D;T) is the pair cd(D;T)= (c(D;T), d(D;T)) together with the dictionary order. d(D;T) ≦ c(D;T) implies: Note(A. Shimizu). The dictionary order on (c(D;T), d(D;T)) is equivalent to the numerical order on c(D;T) 2 +d(D;T).

Definition. A spatial graph G is equivalent to a spatial graph G' if ∃ an orientation-preserving homeomorphism h: R 3 → R 3 such that h(G)=G'. Let [G] be the class of spatial graphs G’ which are equivalent to G.

Equivalence Theorem. G and G' are equivalent if and only if any diagram D of G is deformed into any diagram D’ of G’ by a finite sequence of the generalized Reidemeister moves: ⇔ ⇔ ⇔⇔ ⇔⇔ ⇔⇔ Ｉ ＩV ＩV Ｉ ＩＩＩＩＩＩ V

Let [D G ] be the class of diagrams obtained from a diagram D of G by the generalized Reidemeister moves. [G] ⇔ [D G ] Definition. The complexity of a spatial graph G is γ(G) = min{cd(D;T)| (D;T) ∈ [D G ]} (in the dictionary order). Let γ(G)= (c γ (G), d γ (G)).

Note. This complexity is reducible by a crossing change ⇔ or any splice ⇒ or (1)If d γ (G)>0, then ∃ G’ with γ(G’)< γ(G) by a crossing change. d γ (G)=0 ⇔ G is equivalent to G’ with a monotone diagram D’ with c γ (D’)= c γ (G). (2) If c γ (G)>0, then ∃ G’ with γ(G’)< γ(G) by any splice. c γ (G)=0 ⇔ G is equivalent to a graph in a plane.

Definition. The warping degree of G is : d(G)= min{d(D)| (D ;T) ∈ [D G ]} (d γ (G) is called the γ-warping degree.) Definition. G is unknotted if d(G)= 0, and γ-unknotted if d γ (G)=0. A link L is unknotted ⇔ L is trivial. 4. Warping degree and unknotting number

Lemma. For ∀ given graph Γ, ∃ only finitely many unknotted graphs G of Γ up to equivalences. Lemma. An unknotted connected G is deformed into a maximal tree by a sequence of edge reductions: ⇒ ⇒ Corollary. Every edge or loop of an unknotted G is contained in a trivial constituent knot.

Lemma. An unknotted connected graph G is equivalent to a trivial bouquet of circles after some edge contractions: ⇒ A constituent link of a spatial graph G is a link contained in G.

Conway-Gordon Theorem. (1)Every spatial 6-complete graph K 6 contains a non-trivial constituent link. (2)Every spatial 7-complete graph K 7 contains a non-trivial constituent knot. An unknotted K 6 An unknotted K 7

Let γ( Γ ) = min{γ(G) | G is a spatial graph of Γ }. Definition. A Γ -unknotted graph G is a spatial graph of Γ with γ(G) = γ( Γ ). Note. (1) Let γ( Γ )= (c γ ( Γ ), d γ ( Γ )). Then d γ ( Γ )=0. Γ -unknotted ⇒ γ-unknotted ⇒ unknotted. (2) c γ ( Γ )=0 if and only if Γis a plane graph. (3) A spatial plane graph G is Γ -unknotted ⇔ G is equivalent to a graph in a plane. Definition

Note. ∃ a γ-unknotted, non- Γ -unknotted graph. For example, G = has c γ (G)=2 and is γ-unknotted. G is a plane graph with a Hopf link as a constituent link, which is not Γ -unknotted.

Definition. O={unknotted graphs of Γ }. O ={unknotted graphs with monotone diagrams obtained from ∀ (D ;T) ∈ [D G ] with cd(D;T)=γ(G)}. O Γ ={ Γ -unknotted graphs}. Then O ⊃ O ⊃ O Γ. Definition. The (γ, Γ )-warping degree d (G) of G is: d (G) =d γ (G)+ ρ(O,O Γ ). (ρ denotes the Gordian distance.) By definition, d(G) ≦ d γ (G) ≦ d (G). Γ γ Γ γ G γ G γ G γ Γ γ

Let [D G ] γ = {(D;T) ∈ [D G ] | c(D;T)= ｃ γ (G)}. Definition. The unknotting number u(G), the Γ -unknotting number u Γ (G), the γ -unknotting number u γ (G) and the (γ, Γ )-unknotting number u (G) of G are : u(G) = ρ(G,O), u Γ (G) = ρ(G,O Γ ), u γ (G) = ρ(([D G ] γ,O), u (G) = ρ([D G ] γ,O Γ ). Γ γ Γ γ

Theorem. u(G), u Γ (G), u γ (G), u (G), d(G), d γ (G) and d (G) are mutually distinct topological invariants of G and satisfy: u(G) ≦ u Γ (G) ≦ u (G) ≦ d (G) u γ (G) ≦ d γ (G) u(G) ≦ d(G) ≦ ≦ ≦ ≦ Γ γ Γ γ Γ γ Γ γ

Example 1. For G=, c γ (G)=2 and u(G) = u γ (G) = d(G) = d γ (G)= 0. On the other hand, u Γ (G)= d (G)=1. Γ γ

Lemma. (1) (Tat Sang Fung, M. Ozawa) If K is a knot with d(K)=1, then K is a non-trivial twist knot. = (2) If G is a θ-curve with d(G)=1, then G has a non-trivial twist knot as a constituent knot.

Example 2. (T. S. Fung, M. Ozawa, A. Shimizu) For K=5 2,c γ (K)=5, u(K)= u (K)=d(K)=1< d γ (K)= d (K)=2. Example 3. For K = 6 2, c γ (K)=6, u (K)= u (K)= 1 < d(K)= d (K)=2. In fact, u= u = 1: d= d ≦ 2: By Lemma, d(K) ≧ 2 (K is not any twist knot). Γ γ Γ γ Γ γ Γ γ Γ γ Γ γ

Example 4. (Y. Nakanishi 1983, S. A. Bleiler 1984) K=10 8 has u(K) =u Γ (K)=2< u γ (K)=u (K)=3. Example 5. (Kinoshita’s θ-curve) For G =, c γ (G) = 7 and u(G)= u (G)= 1 < d(G)= d (G)= 2. Γ γ Γ γ Γ γ

Γ γ In fact, u(G)= 1 by knottedness of G and by a crossing change. a based diagram a monotone diagram O γ G=O Γ implies u γ (G)=u (G) and d γ (G)= d (G). Since G is non-trivial and the 3 constituent knots are trivial, d(G) ≧ 2 by Lemma. Hence, if c γ (G)=7, then u γ (G)= u (G)= 1, d γ (G)= d (G)=2. Γ γ Γ γ Γ γ

By the diagram, c γ (G) ≦ 7. We show c γ (G) ≧ 7. By the classification of algebraic tangles with crossing numbers ≦ 6 in: H. Moriuchi, Enumeration of algebraic tangles with applications to theta-curves and handcuff graphs, Kyungpook Math. J.48(2008), The Kinoshita’s θ-curve G cannot have any based diagram with crossing number ≦ 6. Hence c γ (G)=7.

Theorem. For ∀ given graph Γ and any integer n ≧ 0, ∃ infinitely many spatial graphs G of Γ such that u(G)=u Γ (G)= u γ (G)=u (G)=d(G)=d γ (G)=d (G) =n. Proof. Let Γ= the union of a maximal tree T and oriented edges or loops α i (i=1,2,…,s). Let G 0 be a Γ -unknotted graph, and K a trefoil knot. Then d γ (K)= d (K)=1 and Δ K (t) is non-trivial. Let G =G 0 #nK (along an α i ). Then d (G) ≦ n. We show u(G) ≧ n. Let G’ be an unknotted graph obtained from G by k crossing changes. Γ γ Γ γ Γ γ Γ γ

Lemma. Let m(G) and m(G’) be the minimal Λ- generators of the Λ-modules M(G) and M(G’), respectively, associated with the epimorphism H 1 ( R 3 ＼ G) → Z sending the meridian of α i to 1 for ∀ i. Then |m(G)-m(G’)| ≦ k. The proof is in: A. Kawauchi, Distance between links by zero-linking twists, Kobe J.Math.13(1996), Using that π 1 ( R 3 ＼ G’) is a free group of rank s, we have m (G’ )= s-1 and m(G)=s-1+n, for m(K)=1. Hence n ≦ k by Lemma.

5. A spatial graph with degree one vertices Let Γbe a finite polygonal graph with degree 1 vertices. A spatial graph of Γ is an embedding image G of Γ into R 3 such that f(G) is a polygonal graph in R 3 for an orientation-preserving homeomorphism f: R 3 →R 3. Let V be the set of degree 1 vertices of G. For a, b ∈ R 3, let [a,b] be the line segment between a and b. Assume G ≠ [a,b]. For x ∈ G, let P v (x)=[v,x] ∪ ( [v,v’]) be a star with origin v. ∪ v’ ∈ V

Assume G v (x) = G ∪ P v (x) is a spatial graph for every v ∈ V and x ∈ G. Note: It may have v as a degree 1 vertex when V={v}. In this case, we take G v (x) as the spatial graph obtained by contracting the edge with the degree 1 vertex. x v’v v x

Definition. d(G,x)= d(G v (x)) the warping degree of (G,x). u(G,x)= u(G v (x)) the unknotting number of (G,x). For x ∈ V, d(G)= d(G,x) and u(G)= u(G,x) are called the warping degree and the unknotting number of G, respectively. Similarly, u Γ (G,x), u Γ (G), u γ (G,x), u γ (G), u (G,x), u (G), d γ (G,x), d γ (G), d (G,x), d (G) are defined. Different invariants taking min or average in place of max are also defined. max v ∈ V max v ∈ V Γ γ Γ γ Γ γ Γ γ

Example. (1) u(G)=d(G)= u(G,x)=d(G,x)=1 u(G)=d(G)= u(G,x)=d(G,x)=0 u(G)=d(G)=0, u(G,x)=d(G,x)=1 v’ x v x v x v

(2) u(G)=d(G)=0. u(G)=d(G)=0. u(G)=d(G)=1. ● ●