Volume 12 (2007)

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Thin position for knots and 3–manifolds: a unified approach

Hugh Howards, Yo'av Rieck and Jennifer Schultens

Geometry & Topology Monographs 12 (2007) 89–120

DOI: 10.2140/gtm.2007.12.89

References

1

M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987) 237–300 MR881270

2

D Gabai, Foliations and the topology of 3–manifolds III, J. Differential Geom. 26 (1987) 479–536 MR910018

3

H Goda, M Scharlemann, A Thompson, Levelling an unknotting tunnel, Geom. Topol. 4 (2000) 243–275 MR1778174

4

C Hayashi, K Shimokawa, Thin position of a pair (3–manifold, 1–submanifold), Pacific J. Math. 197 (2001) 301–324 MR1815259

5

D J Heath, T Kobayashi, Essential tangle decomposition from thin position of a link, Pacific J. Math. 179 (1997) 101–117 MR1452527

6

D J Heath, T Kobayashi, Locally thin position for a link, Kobe J. Math. 20 (2003) 1–10 MR2102296

7

D J Heath, T Kobayashi, A search method for thin positions of links, Algebr. Geom. Topol. 5 (2005) 1027–1050 MR2171802

8

J Hempel, 3–manifolds, AMS Chelsea Publishing, Providence, RI (2004) MR2098385 (reprint of the 1976 original)

9

H Howards, J Schultens, Thin position for knots and 3–manifolds, preprint

10

W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society, Providence, R.I. (1980) MR565450

11

T Kobayashi, Y Rieck, Knot exteriors with additive Heegaard genus and Morimoto's conjecture arXiv:math/0701765

12

T Kobayashi, Y Rieck, Knots with g(e(k)) = 2 and g(e(k#k#k)) = 6 and Morimoto's conjecture arXiv:math/0701766

13

T Kobayashi, Y Rieck, Heegaard genus of the connected sum of m–small knots, Comm. Anal. Geom. 14 (2006) 1037–1077 MR2287154

14

K Morimoto, On the super additivity of tunnel number of knots, Math. Ann. 317 (2000) 489–508 MR1776114

15

K Morimoto, J Schultens, Tunnel numbers of small knots do not go down under connected sum, Proc. Amer. Math. Soc. 128 (2000) 269–278 MR1641065

16

Y Rieck, Heegaard structures of manifolds in the Dehn filling space, Topology 39 (2000) 619–641 MR1746912

17

Y Rieck, J H Rubinstein, Invariant Heegaard surfaces in manifolds with involutions and the Heegaard genus of double covers arXiv:math/0607145

18

Y Rieck, E Sedgwick, Finiteness results for Heegaard surfaces in surgered manifolds, Comm. Anal. Geom. 9 (2001) 351–367 MR1846207

19

Y Rieck, E Sedgwick, Thin position for a connected sum of small knots, Algebr. Geom. Topol. 2 (2002) 297–309 MR1917054

20

J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3–dimensional manifolds, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 1–20 MR1470718

21

T Saito, M Scharlemann, J Schultens, Lecture notes on generalized Heegaard splittings arXiv:math/0504167

22

M Scharlemann, Thin position in the theory of classical knots, from: "Handbook of knot theory", Elsevier B. V., Amsterdam (2005) 429–459 MR2179267

23

M Scharlemann, J Schultens, Annuli in generalized Heegaard splittings and degeneration of tunnel number, Math. Ann. 317 (2000) 783–820 MR1777119

24

M Scharlemann, J Schultens, 3–manifolds with planar presentations and the width of satellite knots, Trans. Amer. Math. Soc. 358 (2006) 3781–3805 MR2218999

25

M Scharlemann, A Thompson, Thin position and Heegaard splittings of the 3–sphere, J. Differential Geom. 39 (1994) 343–357 MR1267894

26

M Scharlemann, A Thompson, Thin position for 3–manifolds, from: "Geometric topology (Haifa, 1992)", Contemp. Math. 164, Amer. Math. Soc., Providence, RI (1994) 231–238 MR1282766

27

M Scharlemann, A Thompson, On the additivity of knot width, from: "Proceedings of the Casson Fest", Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry (2004) 135–144 MR2172481

28

H Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245–288

29

J Schultens, Bridge numbers of torus knots, to appear in Math. Proc. Cambridge Philos. Soc.

30

J Schultens, The classification of Heegaard splittings for (compact orientable surface) ×  S¹, Proc. London Math. Soc. (3) 67 (1993) 425–448 MR1226608

31

J Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003) 539–544 MR2018265

32

A Thompson, Thin position and bridge number for knots in the 3–sphere, Topology 36 (1997) 505–507 MR1415602

33

M Tomova, Compressing thin spheres in the complement of a link, Topology Appl. 153 (2006) 2987–2999 MR2248402

34

Y-Q Wu, Thin position and essential planar surfaces, Proc. Amer. Math. Soc. 132 (2004) 3417–3421 MR2073319