关于纽结补空间Heegaard分解的稳定化问题
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摘要
Heegaard分解稳定化问题是Heegaard分解理论中重要组成部分.对于一般空间的Heegaard分解寻找稳定化圆片比较困难,但对于纽结补空间T. Kobayashi和T. Saito给出了一种非常有效的方法.本文的主要结论之一便是对他们结果的条件作适当的修改后,得出相似的结果.
M. Scharlemann和J. Schultens指出: n个非平凡纽结连通和的洞数(tunnel number)大于或等于n .作为本文的一个应用,我们给出了一个例子,使得上述等号成立.
一般地,对于纽结的两个解结隧道系统τ和τ′,即使τ和τ′合痕,τ∪τ′也未必还是这个纽结的解结隧道系统.在本文中我们考虑洞数1纽结的这样两个解结隧道系统:它们分别由一条弧和n条弧构成,我们说:当S~3中洞数为1的纽结满足某一特定的条件时,它们具有这种可加性.
The stabilization of Heegaard splitting is one of the most important theories of Heegaard splitting. Generally, it may be difficult to find the reducing disks of a given Heegaard splitting. Fortunately, T. Kobayashi and T. Saito have given a necessary and sufficient condition for a stabilized Heegaard splitting of knot exteriors. In this article, we give a similar theorem based on the T. Kobayashi and T. Saito’s.
M. Scharlemann and J. Schultens proved that the tunnel number of the connected sum of n non-trivial knots is greater or equal to n. As an application, we give an example to support their idea that the equation can hold.
Generally speaking, the union of two unkotting tunnel systems of a knot may be not a unknotting tunnel system, even if they are isotopic. In this article, we consider two unkotting tunnel systems for tunnel number one knot consisting of one and n arcs, respectively, and give a condition such that the union of them is still a unkotting tunnel system.
M. Scharlemann和J. Schultens指出: n个非平凡纽结连通和的洞数(tunnel number)大于或等于n .作为本文的一个应用,我们给出了一个例子,使得上述等号成立.
一般地,对于纽结的两个解结隧道系统τ和τ′,即使τ和τ′合痕,τ∪τ′也未必还是这个纽结的解结隧道系统.在本文中我们考虑洞数1纽结的这样两个解结隧道系统:它们分别由一条弧和n条弧构成,我们说:当S~3中洞数为1的纽结满足某一特定的条件时,它们具有这种可加性.
The stabilization of Heegaard splitting is one of the most important theories of Heegaard splitting. Generally, it may be difficult to find the reducing disks of a given Heegaard splitting. Fortunately, T. Kobayashi and T. Saito have given a necessary and sufficient condition for a stabilized Heegaard splitting of knot exteriors. In this article, we give a similar theorem based on the T. Kobayashi and T. Saito’s.
M. Scharlemann and J. Schultens proved that the tunnel number of the connected sum of n non-trivial knots is greater or equal to n. As an application, we give an example to support their idea that the equation can hold.
Generally speaking, the union of two unkotting tunnel systems of a knot may be not a unknotting tunnel system, even if they are isotopic. In this article, we consider two unkotting tunnel systems for tunnel number one knot consisting of one and n arcs, respectively, and give a condition such that the union of them is still a unkotting tunnel system.
引文
1 J. Milnor. On Manifolds Homeomorphic to the 7-sphere. Ann. of Math.. 1956, 64(2): 399-405
2 J. Milnor. Differentiable Structures on Spheres. Amer. J. Math.. 1959, 81: 962-972
3 W. Jaco. Lectures on 3-Manifold Topology. Amer. Math. Soc.. 1980
4 D. Gabai. Foliations and the Topology of Three Manifolds III. Journal of Differential Geometry. 1987, 26:179~336
5 K. Johannson. Topology and Combinatorics of 3-Manifolds. Lecture Notes in Math.. Springer, 1995
6 W. P. Thurston. Three Dimensional Manifolds. Kleinian Group and Hyperbolic Geometry. Bull. Amer. Math. Soc.. 1982, 6: 412~436
7 J. Hemple. 3-Manifolds. Princeton University Press, 1976
8 F. Waldhausen. On Irreducible 3-Manifolds which are Sufficiently Large. Ann. of Math.. 1968, 87: 56~88
9 W. P. Thurston. The Geometry and Topology of 3-Manifolds. Princeton University Press, 1998
10 H. Doll. A Generalized Bridge Number for Links in 3-Manifolds. Math Ann.. 1992, 294: 701~717
11 D.Rolfsen. Knots and Links. Publish or Perish, 1976
12 K. Morimoto. M. Sakuma. On Unknotting Tunnels for Knots. Math. Ann.. 1991, 289: 143~167
13 T. Kobayashi. Classification of Unknotting Tunnels for Two Bridge Knots. Geometry and Topology Monographs. 1999, 2: 259~290
14 W. B. R. Lickorish. A Representation of Combinatorial 3-Manifolds. Ann.of Math.. 1962, 76: 531~538
15 H. Hilden. Every Closed Orientable 3-Manifold is a 3 Fold Branched Covering Space of 3 Shpere. Bull. Amer. Math. Soc.. 1974, 80: 1243~1244
16 J. Montesions. A Representation of Closed Orientable 3-Manifold as a 3 Fold Branched Covering Space of 3 Shpere. Bull. Amer. Math. Soc.. 1974, 80: 845~846
17 S. Bleiler, Y. Moriah. Heegaard Splittings and Branched Coverings of B3. Math. Ann.. 1988, 281: 531~543
18 M. Lackenby. Classification of Alternating Knot with Tunnel Number One. ArXiv: Math/0103217. Preprint
19 E. E. Moise. Affine Structures in 3-Manifolds V: the Triangulation Theorem and Hauptvermutung. Ann. of Math.. 1952, 55: 96~114
20 R. H. Bing. Necessary and Sufficient Conditions that a 3-Manifold Be S3. Ann. of Math.. 1958, 163(1):17~37
21 P. Heegaard. Forstudier til en Topologiskteori for de Algebraiske Aladers Sammenhaeng. Ph.D.Thesis. 1899
22 Y. Moriah, H. J. Rubinstein. Heegaard Structures of Negatively Curved Manifolds. Communications in Geometrial Annals. 1997, 5(3): 375~412
23 M. Scharlemann, A.Thompson. Thin Position for 3-Manifolds. Comtem-porary Math.. 1994, 164: 231~238
24 K. Johannson. On Surfaces and Heegaard Surfaces. Trans. Amer. Math. Soc.. 1991, 325: 573~591
25 M. Jones, M. Scharlemann. How a Strongly Irreducible Heegaard Splitting Intersects a Handlebody. Topology and its Applications. 2001, 110: 289~301
26 W. Haken. Theorie der Normal Fl?chen. Acta Math.. 1961, 105: 245~275
27 W. Haken. Some Results on Surfaces in 3-Manifolds. Studies in Modern Topology. 1968, 88:34~98
28 M. Ochiai. On Haken’s Theorem and its Extension. Osaka J. Math.. 1983, 20: 461~468
29 A. Casson, C. Gordon. Reducing Heegaard Splittings. Topology and its Applications. 1987, 27: 275~283
30 F. Waldhausen. Heegaard-Zerlegungen der 3-Sphere. Topology. 1968, 7: 195~203
31 F. Lei. On Stability of Heegaard Splittings. Math. Proc. Cambridge Philos. Soc.. 2000, 129: 55~57
32 M. Boileau, M. Rost, H. Zieschang. On Heegaard Decompositions of Torus Knot Exteriors and Related Seifert Fibered Spaces. Math. Ann.. 1998, 279: 553~58
33 M. Scharlemann, Abigail Thompson. The Classification of HeegaardSplittings for (Compact Orientable surfaces)×I. Proc. London. Math. Soc.. 1993, 67: 425~448
34 T. Kobayashi. Heegaard Splittings of Exteriors of Two Bridge Knots. Geometry and Topology. 2001, 5: 609~650
35 J. P. Otal. Sur les Sindements de Heegaard de la Sphere 3. Topology. 1991, 3: 249~258
36 T. Kobayashi, T. Saito. Destabilizing Heegaard Splittings of Knot Exteriors. Preprint
37 J. H. Lee, G. T. Jin. Common stabilizations of Heegaard Splittings of Link Exteriors. Topology and its Applications. 2005, 150: 101~110
38 J. Milnor. A Unique Factorization Theorem for 3-Manifold. Amer. J. Math.. 1962, 84: 1~7
39 J. Singer. Three Dimensional Manifolds and Their Heegaard Diagrams. Trans. Amer. Math. Soc.. 1933, 35(1): 88~111
40 M. Scharlemann, A. Thompson. Heegaard Splittings of (Surface)×I are Standard. Math. Ann.. 1993, 295: 549~564
41 J. Schultens. The Stabilization Problem for Heegaard Splittings of Seifert Fibered Spaces. Topology and its Applications. 1996, 73: 133~139
42 E. Sedgwick. An Infinite Collection of Heegaard Splittings that are Equivalent after One Stabilization. Math. Ann.. 1997, 308: 65~72
43 H. Rubinstein, M. Scharlemann. Comparing Heegaard Splittings of non- Haken 3-Manifolds. Topology. 1996, 35: 1005~1026
44 M. Scharlemann. Heegaard Splittings of Compact 3-Manifolds. Handbook of Geometric Topology. Sherr. North-Holland. Amsterdam. 2002, 921~953
45 M. Scharlemann, T. Saito, J. Schultens. Lecture Notes on Generalized Heegaard Splittings. ArXiv math.GT/0504167. Preprint
46 H. Kneser. Geschlossene Fl?chen in Dreidimensionalen Mannigfaltigkeiten Jahresbericht der Deut. Math.. Verein. 1929, 38 : 248~260
47 M. Scharlemann, J. Schultens. The Tunnel Number of the Connect Sum of n Knots is at Least n. Topology. 1999, 38: 265~27
2 J. Milnor. Differentiable Structures on Spheres. Amer. J. Math.. 1959, 81: 962-972
3 W. Jaco. Lectures on 3-Manifold Topology. Amer. Math. Soc.. 1980
4 D. Gabai. Foliations and the Topology of Three Manifolds III. Journal of Differential Geometry. 1987, 26:179~336
5 K. Johannson. Topology and Combinatorics of 3-Manifolds. Lecture Notes in Math.. Springer, 1995
6 W. P. Thurston. Three Dimensional Manifolds. Kleinian Group and Hyperbolic Geometry. Bull. Amer. Math. Soc.. 1982, 6: 412~436
7 J. Hemple. 3-Manifolds. Princeton University Press, 1976
8 F. Waldhausen. On Irreducible 3-Manifolds which are Sufficiently Large. Ann. of Math.. 1968, 87: 56~88
9 W. P. Thurston. The Geometry and Topology of 3-Manifolds. Princeton University Press, 1998
10 H. Doll. A Generalized Bridge Number for Links in 3-Manifolds. Math Ann.. 1992, 294: 701~717
11 D.Rolfsen. Knots and Links. Publish or Perish, 1976
12 K. Morimoto. M. Sakuma. On Unknotting Tunnels for Knots. Math. Ann.. 1991, 289: 143~167
13 T. Kobayashi. Classification of Unknotting Tunnels for Two Bridge Knots. Geometry and Topology Monographs. 1999, 2: 259~290
14 W. B. R. Lickorish. A Representation of Combinatorial 3-Manifolds. Ann.of Math.. 1962, 76: 531~538
15 H. Hilden. Every Closed Orientable 3-Manifold is a 3 Fold Branched Covering Space of 3 Shpere. Bull. Amer. Math. Soc.. 1974, 80: 1243~1244
16 J. Montesions. A Representation of Closed Orientable 3-Manifold as a 3 Fold Branched Covering Space of 3 Shpere. Bull. Amer. Math. Soc.. 1974, 80: 845~846
17 S. Bleiler, Y. Moriah. Heegaard Splittings and Branched Coverings of B3. Math. Ann.. 1988, 281: 531~543
18 M. Lackenby. Classification of Alternating Knot with Tunnel Number One. ArXiv: Math/0103217. Preprint
19 E. E. Moise. Affine Structures in 3-Manifolds V: the Triangulation Theorem and Hauptvermutung. Ann. of Math.. 1952, 55: 96~114
20 R. H. Bing. Necessary and Sufficient Conditions that a 3-Manifold Be S3. Ann. of Math.. 1958, 163(1):17~37
21 P. Heegaard. Forstudier til en Topologiskteori for de Algebraiske Aladers Sammenhaeng. Ph.D.Thesis. 1899
22 Y. Moriah, H. J. Rubinstein. Heegaard Structures of Negatively Curved Manifolds. Communications in Geometrial Annals. 1997, 5(3): 375~412
23 M. Scharlemann, A.Thompson. Thin Position for 3-Manifolds. Comtem-porary Math.. 1994, 164: 231~238
24 K. Johannson. On Surfaces and Heegaard Surfaces. Trans. Amer. Math. Soc.. 1991, 325: 573~591
25 M. Jones, M. Scharlemann. How a Strongly Irreducible Heegaard Splitting Intersects a Handlebody. Topology and its Applications. 2001, 110: 289~301
26 W. Haken. Theorie der Normal Fl?chen. Acta Math.. 1961, 105: 245~275
27 W. Haken. Some Results on Surfaces in 3-Manifolds. Studies in Modern Topology. 1968, 88:34~98
28 M. Ochiai. On Haken’s Theorem and its Extension. Osaka J. Math.. 1983, 20: 461~468
29 A. Casson, C. Gordon. Reducing Heegaard Splittings. Topology and its Applications. 1987, 27: 275~283
30 F. Waldhausen. Heegaard-Zerlegungen der 3-Sphere. Topology. 1968, 7: 195~203
31 F. Lei. On Stability of Heegaard Splittings. Math. Proc. Cambridge Philos. Soc.. 2000, 129: 55~57
32 M. Boileau, M. Rost, H. Zieschang. On Heegaard Decompositions of Torus Knot Exteriors and Related Seifert Fibered Spaces. Math. Ann.. 1998, 279: 553~58
33 M. Scharlemann, Abigail Thompson. The Classification of HeegaardSplittings for (Compact Orientable surfaces)×I. Proc. London. Math. Soc.. 1993, 67: 425~448
34 T. Kobayashi. Heegaard Splittings of Exteriors of Two Bridge Knots. Geometry and Topology. 2001, 5: 609~650
35 J. P. Otal. Sur les Sindements de Heegaard de la Sphere 3. Topology. 1991, 3: 249~258
36 T. Kobayashi, T. Saito. Destabilizing Heegaard Splittings of Knot Exteriors. Preprint
37 J. H. Lee, G. T. Jin. Common stabilizations of Heegaard Splittings of Link Exteriors. Topology and its Applications. 2005, 150: 101~110
38 J. Milnor. A Unique Factorization Theorem for 3-Manifold. Amer. J. Math.. 1962, 84: 1~7
39 J. Singer. Three Dimensional Manifolds and Their Heegaard Diagrams. Trans. Amer. Math. Soc.. 1933, 35(1): 88~111
40 M. Scharlemann, A. Thompson. Heegaard Splittings of (Surface)×I are Standard. Math. Ann.. 1993, 295: 549~564
41 J. Schultens. The Stabilization Problem for Heegaard Splittings of Seifert Fibered Spaces. Topology and its Applications. 1996, 73: 133~139
42 E. Sedgwick. An Infinite Collection of Heegaard Splittings that are Equivalent after One Stabilization. Math. Ann.. 1997, 308: 65~72
43 H. Rubinstein, M. Scharlemann. Comparing Heegaard Splittings of non- Haken 3-Manifolds. Topology. 1996, 35: 1005~1026
44 M. Scharlemann. Heegaard Splittings of Compact 3-Manifolds. Handbook of Geometric Topology. Sherr. North-Holland. Amsterdam. 2002, 921~953
45 M. Scharlemann, T. Saito, J. Schultens. Lecture Notes on Generalized Heegaard Splittings. ArXiv math.GT/0504167. Preprint
46 H. Kneser. Geschlossene Fl?chen in Dreidimensionalen Mannigfaltigkeiten Jahresbericht der Deut. Math.. Verein. 1929, 38 : 248~260
47 M. Scharlemann, J. Schultens. The Tunnel Number of the Connect Sum of n Knots is at Least n. Topology. 1999, 38: 265~27