| 1 |
J Alexander, A
lemma on systems of knotted curves, Proc. Nat. Acad. Sci.
USA 9 (1963) 103–124 |
| 2 |
D Bachman, D
Cooper, M E White, Large embedded balls
and Heegaard genus in negative curvature, Algebr. Geom.
Topol. 4 (2004) 31–47 MR2031911 |
| 3 |
S A Bleiler, Y
Moriah, Heegaard splittings and
branched coverings of B³, Math. Ann. 281 (1988)
531–543 MR958258 |
| 4 |
M Boileau, J P
Otal, Groupe des difféotopies de certaines
variétés de Seifert, C. R. Acad. Sci. Paris
Sér. I Math. 303 (1986) 19–22 MR849619 |
| 5 |
M Boileau, M
Rost, H Zieschang, On Heegaard
decompositions of torus knot exteriors and related Seifert
fibre spaces, Math. Ann. 279 (1988) 553–581
MR922434 |
| 6 |
M Boileau, H
Zieschang, Heegaard genus of closed
orientable Seifert 3-manifolds, Invent. Math. 76 (1984)
455–468 MR746538 |
| 7 |
G Burde, H
Zieschang, Knots, de Gruyter Studies in Mathematics
5, Walter de Gruyter & Co. (1985) MR808776 |
| 8 |
A J Casson,
C M Gordon, Reducing
Heegaard splittings, Topology Appl. 27 (1987)
275–283 MR918537 |
| 9 |
M Culler, C M
Gordon, J Luecke, P B Shalen,
Dehn surgery on
knots, Ann. of Math. (2) 125 (1987) 237–300
MR881270 |
| 10 |
H Doll, A generalized bridge
number for links in 3–manifolds, Math. Ann. 294
(1992) 701–717 MR1190452 |
| 11 |
M Eudave-Muñoz,
Incompressible
surfaces and (1,1)–knots, J. Knot Theory
Ramifications 15 (2006) 935–948 MR2251034 |
| 12 |
R H Fox, Free
differential calculus I: Derivation in the free group ring,
Ann. of Math. 55 (1953) 547–560 |
| 13 |
K Funcke, Nicht frei
äquivalente Darstellungen von Knotengruppen mit einer
definierierenden Relation, Math. Z. 141 (1975)
205–217 MR0372044 |
| 14 |
C M Gordon, J
Luecke, Knots with
unknotting number 1 and essential Conway spheres,
Algebr. Geom. Topol. 6 (2006) 2051–2116 MR2263059 |
| 15 |
C M Gordon,
A W Reid, Tangle
decompositions of tunnel number one knots and links, J.
Knot Theory Ramifications 4 (1995) 389–409 MR1347361 |
| 16 |
Y Hagiwara,
Reidemeister–Singer distance for unknotting tunnels of
a knot, Kobe J. Math. 11 (1994) 89–100 MR1309994 |
| 17 |
J Hempel,
3–manifolds, Annals of Mathematics Studies 86,
Princeton University Press (1976) |
| 18 |
J Hempel, 3–manifolds
as viewed from the curve complex, Topology 40 (2001)
631–657 MR1838999 |
| 19 |
W Jaco, Lectures on
three-manifold topology, CBMS Regional Conference Series in
Mathematics 43, American Mathematical Society (1980) MR565450 |
| 20 |
K Johannson,
Heegaard surfaces in Haken 3–manifolds, Bull.
Amer. Math. Soc. (N.S.) 23 (1990) 91–98 MR1027902 |
| 21 |
K Johannson,
Topology and combinatorics of 3–manifolds, Lecture
Notes in Mathematics 1599, Springer (1995) MR1439249 |
| 22 |
J Johnson, A
Thompson, On tunnel number one knots which are not
(1,n), arXiv:math.GT/0606226 |
| 23 |
I Kapovich, R
Weidmann, Kleinian groups and the rank problem,
Geom. Topol. (to appear) |
| 24 |
T Kobayashi, There
exist 3-manifolds with arbitrarily high genus irreducible
Heegaard splittings, unpublished |
| 25 |
T Kobayashi,
Structures of the Haken manifolds with Heegaard splittings
of genus two, Osaka J. Math. 21 (1984) 437–455
MR752472 |
| 26 |
T Kobayashi, A
criterion for detecting inequivalent tunnels for a knot,
Math. Proc. Cambridge Philos. Soc. 107 (1990) 483–491
MR1041480 |
| 27 |
T Kobayashi, A
construction of 3–manifolds whose homeomorphism classes
of Heegaard splittings have polynomial growth, Osaka J.
Math. 29 (1992) 653–674 MR1192734 |
| 28 |
T Kobayashi,
A
construction of arbitrarily high degeneration of tunnel numbers
of knots under connected sum, J. Knot Theory
Ramifications 3 (1994) 179–186 MR1279920 |
| 29 |
T Kobayashi,
Heegaard
splittings of exteriors of two bridge knots, Geom.
Topol. 5 (2001) 609–650 MR1857522 |
| 30 |
T Kobayashi, Y
Rieck, Knot exteriors with additive Heegaard genus and
Morimoto's Conjecture, arXiv:math.GT/0701765 |
| 31 |
T Kobayashi, Y
Rieck, Knots with g(E(K))=2 and g(E(K#K#K))=6 and
Morimoto's Conjecture, arXiv:math.GT/070176 |
| 32 |
T Kobayashi, Y
Rieck, Manifolds admitting both strongly irreducible and
weakly reducible Heegard splittings, preprint |
| 33 |
T Kobayashi, Y
Rieck, On the growth rate of tunnel number of knots,
arXiv:math.GT/0402025 |
| 34 |
T Kobayashi, Y
Rieck, Heegaard genus of the connected sum of
m–small knots, Comm. Anal. Geom. (to appear) arXiv:math.GT/0503229 |
| 35 |
M Lustig, Y
Moriah, Nielsen
equivalence in Fuchsian groups and Seifert fibered
spaces, Topology 30 (1991) 191–204 MR1098913 |
| 36 |
M Lustig, Y
Moriah, Generalized Montesinos
knots, tunnels and N–torsion, Math. Ann. 295
(1993) 167–189 MR1198847 |
| 37 |
M Lustig, Y
Moriah, Generating systems of groups and
Reidemeister–Whitehead torsion, J. Algebra 157 (1993)
170–198 |
| 38 |
M Lustig, Y
Moriah, On the complexity of the
Heegaard structure of hyperbolic 3-manifolds, Math. Z.
226 (1997) 349–358 MR1483536 |
| 39 |
M Lustig, Y
Moriah, Closed
incompressible surfaces in complements of wide knots and
links, Topology Appl. 92 (1999) 1–13 MR1670164 |
| 40 |
M Lustig, Y
Moriah, 3–manifolds
with irreducible Heegaard splittings of high genus,
Topology 39 (2000) 589–618 MR1746911 |
| 41 |
M Lustig, Y
Moriah, A finiteness
result for Heegaard splittings, Topology 43 (2004)
1165–1182 MR2079999 |
| 42 |
R C Lyndon,
P E Schupp, Combinatorial group theory,
Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer
(1977) MR0577064 |
| 43 |
Y Minsky, Y
Moriah, S Schleimer, Knots of high distance,
preprint |
| 44 |
Y Moriah, Heegaard
splittings and group presentations, PhD thesis, University
of Texas at Austin (1986) |
| 45 |
Y Moriah, Heegaard splittings of
Seifert fibered spaces, Invent. Math. 91 (1988)
465–481 MR928492 |
| 46 |
Y Moriah, On boundary
primitive manifolds and a theorem of
Casson–Gordon, Topology Appl. 125 (2002)
571–579 MR1935173 |
| 47 |
Y Moriah, Connected
sums of knots and weakly reducible Heegaard splittings,
Topology Appl. 141 (2004) 1–20 MR2058679 |
| 48 |
Y Moriah, H
Rubinstein, Heegaard structures of negatively curved
3–manifolds, Comm. Anal. Geom. 5 (1997) 375–412
MR1487722 |
| 49 |
Y Moriah, S
Schleimer, E Sedgwick, Heegaard
splittings of the form H+nK, Comm. Anal. Geom. 14
(2006) 215–247 MR2255010 |
| 50 |
Y Moriah, J
Schultens, Irreducible
Heegaard splittings of Seifert fibered spaces are either
vertical or horizontal, Topology 37 (1998)
1089–1112 MR1650355 |
| 51 |
Y Moriah, E
Sedgwick, Closed essential
surfaces and weakly reducible Heegaard splittings in manifolds
with boundary, J. Knot Theory Ramifications 13 (2004)
829–843 MR2088748 |
| 52 |
K Morimoto,
On
the additivity of tunnel number of knots, Topology
Appl. 53 (1993) 37–66 MR1243869 |
| 53 |
K Morimoto,
On
composite tunnel number one links, Topology Appl. 59
(1994) 59–71 MR1293117 |
| 54 |
K Morimoto,
There are knots
whose tunnel numbers go down under connected sum, Proc.
Amer. Math. Soc. 123 (1995) 3527–3532 MR1317043 |
| 55 |
K Morimoto,
On the super
additivity of tunnel number of knots, Math. Ann. 317
(2000) 489–508 MR1776114 |
| 56 |
K Morimoto,
Tunnel
number, connected sum and meridional essential
surfaces, Topology 39 (2000) 469–485 MR1746903 |
| 57 |
K Morimoto, M
Sakuma, On unknotting tunnels
for knots, Math. Ann. 289 (1991) 143–167 MR1087243 |
| 58 |
K Morimoto, M
Sakuma, Y Yokota, Examples of tunnel number one
knots which have the property “1+1=3”, Math.
Proc. Cambridge Philos. Soc. 119 (1996) 113–118 MR1356163 |
| 59 |
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 |
| 60 |
J Nielsen, Die Isomorphismen der
allgemeinen, unendlichen Gruppe mit zwei Erzeugenden,
Math. Ann. 78 (1964) 385–397 MR1511907 |
| 61 |
F H Norwood,
Every
two-generator knot is prime, Proc. Amer. Math. Soc. 86
(1982) 143–147 MR663884 |
| 62 |
P Ozsváth, Z
Szabó, Knots with
unknotting number one and Heegaard Floer homology,
Topology 44 (2005) 705–745 MR2136532 |
| 63 |
D Rolfsen, Knots
and links, Mathematics Lecture Series 7, Publish or Perish
(1976) MR0515288 |
| 64 |
H Rubinstein, M
Scharlemann, Comparing
Heegaard splittings of non-Haken 3–manifolds,
Topology 35 (1996) 1005–1026 MR1404921 |
| 65 |
M Sakuma, Manifolds
with infinitely many non-isotopic Heegaard splittings of
minimal genus, from: "(Unofficial) Proceedings of the
Conference on Various Structures on Knots and their
Applications (Osaka City University)" (1988) 172–179 |
| 66 |
M Scharlemann,
There are no
unexpected tunnel number one knots of genus one, Trans.
Amer. Math. Soc. 356 (2004) 1385–1442 MR2034312 |
| 67 |
M Scharlemann, J
Schultens, The tunnel
number of the sum of n knots is at least n, Topology 38
(1999) 265–270 MR1660345 |
| 68 |
M Scharlemann, J
Schultens, Comparing
Heegaard and JSJ structures of orientable
3–manifolds, Trans. Amer. Math. Soc. 353 (2001)
557–584 MR1804508 |
| 69 |
M Scharlemann, A
Thompson, Heegaard splittings of (\textsurface)×I
are standard, Math. Ann. 295 (1993) 549–564 |
| 70 |
M Scharlemann, M
Tomova, Alternate Heegaard
genus bounds distance, Geom. Topol. 10 (2006)
593–617 MR2224466 |
| 71 |
H Schubert, Knoten
mit zwei Brücken, Math. Z. 65 (1956)
133–170 |
| 72 |
J Schultens, The
classification of Heegaard splittings for (compact orientable
surface) × S1, Proc. London Math. Soc. (3)
67 (1993) 425–448 MR1226608 |
| 73 |
J Schultens,
Additivity
of tunnel number for small knots, Comment. Math. Helv.
75 (2000) 353–367 MR1793793 |
| 74 |
J Schultens, R
Weidmann, On the geometric and algebraic rank of graph
manifolds, preprint |
| 75 |
E Sedgwick,
An
infinite collection of Heegaard splittings that are equivalent
after one stabilization, Math. Ann. 308 (1997)
65–72 MR1446199 |
| 76 |
E Sedgwick, The
irreducibility of Heegaard splittings of Seifert fibered
spaces, Pacific J. Math. 190 (1999) 173–199 MR1722770 |
| 77 |
E Sedgwick,
Genus two
3–manifolds are built from handle number one
pieces, Algebr. Geom. Topol. 1 (2001)
763–790 |
| 78 |
J Souto, A note on the
tameness of hyperbolic 3–manifolds, Topology 44
(2005) 459–474 MR2114957 |
| 79 |
M Tomova, Multiple
bridge surfaces restrict knot distance, arXiv:math.GT/0511139 |
