COMPLEXITY OF EMBEDDINGS OF A GRAPH INTO THE 3-SPHERE

MAKOTO OZAWA

Abstract. Several complexity of spatial graphs are defined from the 3-manifold theoretic viewpoint, and the relationship between complexities are studied.

1. Definition of complexities of spatial graphs

Let G = (V, E) be a connected graph without degree one vertices and S(G) the set of all embeddings f : G → S^{3} of G into S^{3}.

Definition 1.1. (genus of a graph) g(G) = min{g(F)| there exists an embedding of G into a closed orientable surface F }

Definition 1.2. (genus of a spatial graph) g(f) = min{g(F)| there exists a Heegaard surface F of S^{3} such that F ⊃ f(G)}

Definition 1.3. (tunnel number of a spatial graph) Put t(f) = g(E(f(G))) − β_{1}(G).

Definition 1.4. (trivial spatial graph) f is trivial if g(f) = g(G). Put T (G) = {f ∈ S(G)|f is trivial}.

Definition 1.5. (free spatial graph) f is free if π_{1}(S^{3} − f(G)) is a free group. Put F(G) = {f ∈ S(G)|f is free}

Definition 1.6. (totally knotted spatial graph) f is totally knotted if ∂E(f(G)) is incompressible in E(f(G)). Put T K(G) = {f ∈ S(G)|f is totally knotted}.

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Definition 1.7. (critically trivial, free, totally knotted spatial graph) f is critically trivial (resp. critically free, critically totally knotted) if

1. f is trivial (resp. free, totally knotted)

2. f|_{G−e} is non-trivial (resp. non-free, non-totally knotted) for any e ∈ E. Put

• T_{∗}(G) = {f ∈ S(G)|f is critically trivial}

• F_{∗}(G) = {f ∈ S(G)|f is critically free}

• T K_{∗}(G) = {f ∈ S(G)|f is critically totally knotted} Definition 1.8. (almost trivial, free, totally knotted spatial graph) f is almost trivial if

Key words and phrases. spatinal graph, trivial, free, prime, essential, minimally knotted, totally knotted.

The author was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.

1. f is non-trivial (resp. non-free, non-totally knotted)

2. f|_{G−e} is trivial (resp. free, totally knotted) for any e ∈ E Put

• T^{∗}(G) = {f ∈ S(G)|f is almost trivial}

• F^{∗}(G) = {f ∈ S(G)|f is almost free}

• T K^{∗}(G) = {f ∈ S(G)|f is almost totally knotted}

2. Decomposition of spatial graphs

Definition 2.1. (prime spatial graph) Let S be a 2-sphere intersecting f(G) transversely in the interior of edges. Then S decomposes the pair (S^{3}, f(G)) into two “graph-tangles” (B_{1}, f(G) ∩ B_{1}) and (B_{2}, f(G) ∩ B_{2}). We consider the union of (B_{i}, f(G) ∩ B_{i}) and a standard ball-star pair, and obtain two spatial graphs (S^{3}, f_{i}(G_{i})) (i = 1, 2). Then we denote such decomposition of a spatial graph by f(G) = f_{1}(G_{1})#_{S}f_{2}(G_{2}). A decomposing sphere S is essential if S ∩ E(f(G)) is incompressible and not boundary parallel in E(f(G)).

f is composite if

1. there exists an essential decomposing sphere S for f(G)

2. both f_{1} and f_{2} are non-trivial f is prime if it is not composite. Put P(G) = {f ∈ S(G)|f is prime}.

3. Relationship between complexies

Theorem 3.1. T (G) ≠ ∅.

Theorem 3.2. ([11]) Suppose that g(G) = 0. Then f ∈ T (G) iff 1. f|_{G−e} is trivial for any e ∈ E

2. f is free

Conjecture 3.3. Suppose that g(G) ≥ 1. Then f ∈ T (G) iff 1. g(f|_{G−e}) ≤ g(G) for any e ∈ E

2. f is free

Corollary 3.4. Suppose that g(G) = 0. Then T^{∗}(G) ∩ F(G) = ∅.

Theorem 3.5. T (G) ⊂ F(G).

Corollary 3.6. F(G) ≠ ∅.

Theorem 3.7. T K(G) ≠ ∅.

Theorem 3.8. ([3], [16], [12]) Suppose that g(G) = 0. Then T^{∗}(G) ≠ ∅.

Theorem 3.9. Suppose that g(G − e) = g(G) for any e ∈ E. Then T^{∗}(G) ≠ ∅.

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Theorem 3.10. T (G) = F(G) iff G is a cycle.

Theorem 3.11. ([9]) The following are equivalent.

1. f is free.

2. E(f(G)) is a handlebody.

3. E(f(G)) does not contain an incompressible closed surface.

Corollary 3.12. F(G) ∩ T K(G) = ∅.

Problem 3.13. F(G) ⊂ P(G)?

Theorem 3.14. ([2], [5]) Let G be a theta-curve. Then F(G) ⊂ P(G).

Conjecture 3.15. T^{∗}(G) ⊂ T K(G).

Theorem 3.16. ([13]) Let G be a theta-curve. Then T^{∗}(G) ⊂ T K(G).

Conjecture 3.17. Suppose that f_{1}, f_{2} ∈ T K(G). Then f_{1} and f_{2} are equivalent iff E(f_{1}(G)) and E(f_{2}(G)) are homeomorphic.

4. Proofs

Proof. (of T (G) ≠ ∅) Let F be a closed orientable surface of genus g(G). Then there exists an embedding f_{0} : G → F. Let f_{1} : F → S^{3} be an embedding such that f_{1}(F) is a Heegaard surface of S^{3}. Then f_{1}|_{G} ∘ f_{0} is trivial.

Proof. (of T (G) ⊂ F(G)) Let f ∈ T (G) and F be a Heegaard surface of genus g(G) containing f(G). Then each component of F − f(G) is an open disk. Otherwise, there exists an embedding f_{0} : G → F′ such that F′ is a closed orientable surface of genus less than g(G), a contradiction. Hence E(f(G)) is homeomorphic to (genus g(G) handlebody)∪(neighborhood of disks F ∩E(f(G))∪(genus g(G) handlebody), and E(f(G)) is a handlebody. Therefore π_{1}(S^{3} − f(G)) is a free group.

Proof. (of T K(G) ≠ ∅) Let T be a spanning tree of G and N(T) be a regular neighborhood of f_{0}(T) for some f_{0} ∈ S(G). Recall that we have a tangle whose exterior is boundary-irreducible (see for example, [17]). Let (B, S) be a β_{1}(G)-string boundary-irreducible tangle, and put (S^{3}, f(G)) = (N(T), f_{0}(G) ∩ N(T)) ∪ (B, S). Then we have f ∈ T K(G).

Proof. (of 3.9) Let f ∈ T (G). Note that f|_{G−e} is trivial for any e ∈ E. Then the proof follows the Proof of [12, Theorem 1].

Proof. (of 3.10)

Lemma 4.1. Let 3_{1} be a trefoil knot. Then t(#_{i=1}^{n}3_{1}) = n.

Proof. It is easy to see t(#_{i=1}^{n}3_{1}) ≤ n. On the other hand, t(#_{i=1}^{n}3_{1}) ≥ n follows [7], [8], or [10].

Lemma 4.2. (C. Morin - M. Saito’s inequality, c.f. [6], [4, Proposition 15.3.9])

t(f) ≤ g(f) ≤ t(f) + β_{1}(G)

Lemma 4.3. Let G be a cycle and f ∈ S(G) be a trefoil knot. Then g(#_{i=1}^{n}f) = n.

Proof. It is easy to see g(#_{i=1}^{n}f) ≤ n. On the other hand, g(#_{i=1}^{n}f) ≥ n follows 4.1 and 4.2.

Lemma 4.4. There exists f ∈ F(G) such that β_{1}(G) − 1 ≤ g(f).

Proof. Let K = #_{i=1}^{β_{1}(G)−1}3_{1} and U = t_{1} ∪ ⋯ ∪ t_{β_{1}(G)−1} be an unknotting tunnel system for K. By an isotopy of U, we may assume that K ∩ U is one point p. Put B = S^{3} −intN(p). Then the exterior of the tangle (B, (K ∪U)∩B) is a handlebody. Let T be a spanning tree of G and N(T) be a regular neighborhood of f_{0}(T) for some f_{0} ∈ S(G). Put (S^{3}, f(G)) = (N(T), f_{0}(G) ∩ N(T)) ∪ (B, (K ∪ U) ∩ B) and hence f ∈ F(G). Then, since G contains a cycle C such that f(C) represents K, we have g(f) ≥ g(f|_{C}) ≥ β_{1}(G) − 1.

Lemma 4.5. 2g(G) ≤ β_{1}(G).

Proof. Let f ∈ T (G) and r denote the number of regions F − f(G), where F is a Heegaard surface of S^{3} of genus g(G) which contains f(G). By the Euler formula for F,

2 − 2g(G) = 1 − β_{1}(G) + r ≥ 2 − β_{1}

Hence we have 2g(G) ≤ β_{1}(G).

Lemma 4.6. If g(G) ≤ 1, then β_{1}(G) ≥ 4.

Proof. If G is not planar graph, then by Kuratowski’s theorem, G contains a subdivision of K_{5} or K_{3,3}. Since β_{1}(K_{5}) = 6 and β_{1}(K_{3,3}) = 4, β_{1}(G) ≥ 4.

If g(G) ≥ 2, then by 4.4, there exists f ∈ F(G) such that β_{1}(G) − 1 ≤ g(f). By 4.5,

2g(G) − 1 ≤ β_{1}(G) − 1 ≤ g(f)
1 ≤ g(G) − 1 ≤ g(f) − g(G)

Thus, T (G) ≠ F(G).

Next, if g(G) = 1, then by 4.6 and 4.4, there exists f ∈ F(G) such that

3 ≤ β_{1}(G) − 1 ≤ g(f)

Finally, if g(G) = 0 and β_{1}(G) ≥ 2, then by 4.4, there exists f ∈ F(G) such that g(f) ≥ 1. Otherwise, G must be a cycle.

Conversely, if G is a cycle, then f(G) is the trivial knot for any f ∈ F(G). Thus F(G) ⊂ T (G).

References

1. C. McA. Gordon, On primitive sets of loops in the boundary of a handlebody, Topology Appl. 27 (1987) 285–299.

2. C. McA. Gordon and A. Reid, Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications 4 (1995) 389–409.

3. A. Kawauchi, Almost identical imitations of (3, 1)-dimensional manifold pairs, Osaka J. Math. 26 (1989) 743–758.

4. A. Kawauchi, A survey of Knot Theory, Birkhäuser Verlag, 1996.

5. K. Morimoto, Planar surfaces in a handlebody and a theorem of Gordon-Reid, in Proc. Knots ’96, ed. S. Suzuki, World Sci. Publ. Co., Singapore (1997) 127–146.

6. K. Morimoto, On the additivity of h-genus of knots, Osaka J. Math. 31 (1994) 137–145.

7. K. Morimoto, Tunnel number, connected sum and meridional essential surfaces, Topology 39 (2000) 469–485.

8. K. Morimoto and J. Schultens, Tunnel numbers of small knots do not go down under connected sum, Proc. Amer. Math. Soc.

9. M. Ozawa, Synchronism of an incompressible non-free Seifert surface for a knot and an algebraically split closed surface in the knot complement, Proc. Amer. Math. Soc. 128 (2000) 919–922.

10. M. Scharlemann and J. Schultens, The tunnel number of the connected sum of n knots is at least n, Topology.

11. M. Scharlemann and A. Thompson, Detecting unknotted graphs in 3-space, J. Diff. Geom. 34 (1991) 539–560.

12. T. Soma, Spatial-graph isotopy for trivalent graphs and minimally knotted embeddings, Topology Appl. 73 (1996) 23–41.

13. T. Soma, Re: almost unknotted theta-curve, Knot Mailing List in Japan, (2000) [knot:0491], [knot:0508], [knot:0515].

14. K. Taniyama, Cobordism, homotopy and homology of graphs in ℝ^{3}, Topology 33 (1994) 509– 523.

15. Y.-Q. Wu, On planarity of graphs in 3-manifolds, Comment. Math. Helvetici 67 (1992) 635– 647.

16. Y.-Q. Wu, Minimally knotted embeddings of planar graphs, Math. Z. 214 (1993) 653–658.

17. Y.-Q. Wu,Incompressible surfaces in link complements, preprint available in http://www.math.uiowa.edu/˜ wu/papers/papers.html

Department of Mathematics, School of Education, Waseda University, Nishiwaseda 1-6-1, Shinjuku-ku, Tokyo 169-8050, Japan

   E-mail address: ozawa@musubime.com