A. A. Thompson :
Property \( \mathrm{P} \) for certain classes of knots (genus one, band-connect sum) .
Ph.D. thesis ,
Rutgers University ,
1986 .
Advised by M. G. Scharlemann and J. L. Shaneson .
An article based on this dissertation was published in Topology 26 :2 (1987) .
MR
2634826
phdthesis
People
BibTeX
@phdthesis {key2634826m,
AUTHOR = {Thompson, Abigail A.},
TITLE = {Property \$\mathrm{P}\$ for certain classes
of knots (genus one, band-connect sum)},
SCHOOL = {Rutgers University},
YEAR = {1986},
PAGES = {33},
URL = {https://search.proquest.com/docview/303524889},
NOTE = {Advised by M. G. Scharlemann
and J. L. Shaneson. An article
based on this dissertation was published
in \textit{Topology} \textbf{26}:2 (1987).
MR:2634826.},
}
M. Scharlemann and A. Thompson :
“Finding disjoint Seifert surfaces ,”
Bull. London Math. Soc.
20 : 1
(January 1988 ),
pp. 61–64 .
MR
916076
Zbl
0654.57005
article
Abstract
People
BibTeX
Given two Seifert surfaces \( S \) and \( T \) for a knot \( K \) , there is a sequence of Seifert surfaces \( S = S_0 \) , \( S_1,\dots \) , \( S_n=T \) such that for each \( i \) , \( 1\leq i\leq n \) , \( S_i \) is disjoint from \( S_{i-1} \) . The standard proof (see, for example [4]), which is useful in showing that any two Seifert matrices of \( K \) are \( S \) -equivalent, puts no limit on the genus of the intermediate Seifert surfaces \( S_1,\dots \) , \( S_{n-1} \) . Here we present a simple proof that
if \( S \) and \( T \) are of minimal genus, then we may take all \( S_i \) to be of minimal genus, and
for \( S \) an arbitrary Seifert surface, there is a sequence \( S = S_0 \) , \( S_1,\dots \) , \( S_n \) or Seifert surfaces such that
\[ \operatorname{genus}(S_{i-1}) > \operatorname{genus}(S_i) ,\]
\( S_n \) is of minimal genus, and for each \( i \) , \( 1\leq i \leq n \) ,
\[ S_i \cap S_{i-1} = \emptyset .\]
@article {key916076m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Finding disjoint {S}eifert surfaces},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical
Society},
VOLUME = {20},
NUMBER = {1},
MONTH = {January},
YEAR = {1988},
PAGES = {61--64},
DOI = {10.1112/blms/20.1.61},
NOTE = {MR:916076. Zbl:0654.57005.},
ISSN = {0024-6093},
}
M. Scharlemann and A. Thompson :
“Unknotting number, genus, and companion tori ,”
Math. Ann.
280 : 2
(March 1988 ),
pp. 191–205 .
MR
929535
Zbl
0616.57003
article
Abstract
People
BibTeX
In [Scharlemann 1985a] a complicated combinatorial argument showed that the band sum of knots is unknotted if and only if the band sum is a connected sum of unknots. This argument has since been dramatically simplified [Thompson 1987] and extended [Gabai 1987; S3, Sect. 8] using the newly developed machinery of Gabai. In [Scharlemann 1985b] a similar but more complicated combinatorial argument demonstrated that unknotting number one knots are prime. It seems natural to ask whether the Gabai machinery can simplify the proof of this old conjecture as well.
In fact the Gabai machine reveals a connection between the unknotting number of a knot, its genus, and the position of its companion tori. In Sect. 3 we show (roughly) that, if a single crossing change made to a knot \( K \) reduces its genus by more than one, then any companion torus to \( K \) can be made disjoint from the crossing. In particular, of \( K \) were composite and of unknotting number one, then the swallow-follow companion torus would remain as a companion to the unknot, which is impossible. Therefore no composite knot has unknotting number one.
This argument exploits the drop (by at least two) in the genus of a composite knot when a crossing change unknots it. It is natural to ask whether, in general, the genus of a knot drops (or at least does not rise) as it is unknotted by crossing changes. Knots exist for which a crossing change both lowers the unknotting number and raises the genus. A specific example (due to Chuck Livingston) is given in the appendix. Boileau and Murakami have shown us others. In Sect. 1 we give a general construction, again using the Gabai machine, which seems to produce myriads of examples.
Section 2 is a technical section which readies the Gabai machine (as presented in [Scharlemann 1989]) for use in Sect. 3. In Sect. 4 we view crossing changes as a special case of a more general operation, that of attaching an \( n \) -half-twisted band, and discuss how the main results of Sect. 3 generalize.
@article {key929535m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Unknotting number, genus, and companion
tori},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {280},
NUMBER = {2},
MONTH = {March},
YEAR = {1988},
PAGES = {191--205},
DOI = {10.1007/BF01456051},
NOTE = {MR:929535. Zbl:0616.57003.},
ISSN = {0025-5831},
}
M. Scharlemann and A. Thompson :
“Link genus and the Conway moves ,”
Comment. Math. Helv.
64 : 4
(1989 ),
pp. 527–535 .
MR
1022995
Zbl
0693.57004
article
People
BibTeX
@article {key1022995m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Link genus and the {C}onway moves},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {64},
NUMBER = {4},
YEAR = {1989},
PAGES = {527--535},
DOI = {10.1007/BF02564693},
NOTE = {MR:1022995. Zbl:0693.57004.},
ISSN = {0010-2571},
}
M. Scharlemann and A. Thompson :
“Detecting unknotted graphs in 3-space ,”
J. Diff. Geom.
34 : 2
(1991 ),
pp. 539–560 .
MR
1131443
Zbl
0751.05033
article
Abstract
People
BibTeX
A finite graph \( \Gamma \) is abstractly planar if it is homeomorphic to a graph lying in \( S^2 \) . A finite graph \( \Gamma \) imbedded in \( S^3 \) is planar if \( \Gamma \) lies on an embedded surface in \( S^3 \) which is homeomorphic to \( S^2 \) .
In this paper we give necessary and sufficient conditions for a finite graph \( \Gamma \) in \( S^3 \) to be planar. (All imbeddings will be tame, e.g., PL or smooth.) This can be viewed as an unknotting theorem in the spirit of Papakyriakopolous [1957]: a simple closed curve in \( S^3 \) is unknotted if and only if its complement has free fundamental group.
@article {key1131443m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Detecting unknotted graphs in 3-space},
JOURNAL = {J. Diff. Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {34},
NUMBER = {2},
YEAR = {1991},
PAGES = {539--560},
DOI = {10.4310/jdg/1214447220},
NOTE = {MR:1131443. Zbl:0751.05033.},
ISSN = {0022-040X},
}
M. Scharlemann and A. Thompson :
“Heegaard splittings of \( (\textrm{surface})\times I \) are standard ,”
Math. Ann.
295 : 3
(1993 ),
pp. 549–564 .
MR
1204837
Zbl
0814.57010
article
Abstract
People
BibTeX
Frohman and Hass have shown [1989] that genus three Heegaard splittings of the 3-torus are standard. Boileau and Otal [1990] generalize this result to show that all Heegaard splittings of the 3-torus are standard. A crucial part of Boileau–Otal’s argument is to show that all Heegaard splittings of a torus crossed with an interval are standard. We generalize this part of their paper to prove that all Heegaard splittings of a closed orientable genus \( g \) surface crossed with an interval are standard. Many of our arguments are based on theirs; we differ substantially in Sect. 4, which allows us to obtain the more general result.
The paper is organized as follows: Section 1 begins with a discussion of compression bodies and their spines. In Sect. 2 we discuss Heegaard splittings and state the main theorem. The proof of the main theorem begins in Sect. 3, where we prove a lemma which splits the remainder of the proof into two cases. These cases are considered in Sect. 4 and Sect. 5. In Sect. 6 we exploit the main theorem to generalize a theorem of Frohman.
@article {key1204837m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Heegaard splittings of \$(\textrm{surface})\times
I\$ are standard},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {295},
NUMBER = {3},
YEAR = {1993},
PAGES = {549--564},
DOI = {10.1007/BF01444902},
NOTE = {MR:1204837. Zbl:0814.57010.},
ISSN = {0025-5831},
}
M. Scharlemann and A. Thompson :
“Thin position and Heegaard splittings of the 3-sphere ,”
J. Diff. Geom.
39 : 2
(1994 ),
pp. 343–357 .
MR
1267894
Zbl
0820.57005
article
Abstract
People
BibTeX
We present here a simplified proof of the theorem, originally due to Waldhausen [1968], that a Heegaard splitting of \( S^3 \) is determined solely by its genus. The proof combines Gabai’s powerful idea of “thin position” [1987] with Johannson’s [1991] elementary proof of Haken’s theorem [1968] (Heegaard splittings of reducible 3-manifolds are reducible). In §3.1, 3.2 & 3.8 we borrow from Otal [1991] the idea of viewing the Heegaard splitting as a graph in 3-space in which we seek an unknotted cycle.
Along the way we show also that Heegaard splittings of boundary reducible 3-manifolds are boundary reducible [Casson and Gordon 1987, 1.2], obtain some (apparently new) characterizations of graphs in 3-space with boundary-reducible complement, and recapture a critical lemma of [Menasco and Thompson 1989].
@article {key1267894m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Thin position and {H}eegaard splittings
of the 3-sphere},
JOURNAL = {J. Diff. Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {39},
NUMBER = {2},
YEAR = {1994},
PAGES = {343--357},
DOI = {10.4310/jdg/1214454875},
NOTE = {MR:1267894. Zbl:0820.57005.},
ISSN = {0022-040X},
}
M. Scharlemann and A. Thompson :
“Thin position for 3-manifolds ,”
pp. 231–238
in
Geometric topology
(Haifa, Israel, 10–16 June 1992 ).
Edited by C. Gordon, Y. Moriah, and B. Wajnryb .
Contemporary Mathematics 164 .
American Mathematical Society (Providence, RI ),
1994 .
MR
1282766
Zbl
0818.57013
incollection
Abstract
People
BibTeX
We define thin position for 3-manifolds, and examine its relation to Heegaard genus and essential surfaces in the manifold. We show that if the width of a manifold is smaller than its Heegaard genus then the manifold contains an essential surface of genus less than the Heegard genus.
@incollection {key1282766m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Thin position for 3-manifolds},
BOOKTITLE = {Geometric topology},
EDITOR = {Gordon, Cameron and Moriah, Yoav and
Wajnryb, Bronislaw},
SERIES = {Contemporary Mathematics},
NUMBER = {164},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1994},
PAGES = {231--238},
DOI = {10.1090/conm/164/01596},
NOTE = {(Haifa, Israel, 10--16 June 1992). MR:1282766.
Zbl:0818.57013.},
ISSN = {0271-4132},
ISBN = {9780821851821},
}
M. Scharlemann and A. Thompson :
“Pushing arcs and graphs around in handlebodies ,”
pp. 163–171
in
Low-dimensional topology .
Edited by K. Johannson .
Conference Proceedings and Lecture Notes in Geometry and Topology 3 .
International Press (Cambridge, MA ),
1994 .
MR
1316180
Zbl
0868.57024
incollection
People
BibTeX
@incollection {key1316180m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Pushing arcs and graphs around in handlebodies},
BOOKTITLE = {Low-dimensional topology},
EDITOR = {Johannson, Klaus},
SERIES = {Conference Proceedings and Lecture Notes
in Geometry and Topology},
NUMBER = {3},
PUBLISHER = {International Press},
ADDRESS = {Cambridge, MA},
YEAR = {1994},
PAGES = {163--171},
NOTE = {MR:1316180. Zbl:0868.57024.},
ISBN = {9781571460189},
}
H. Goda, M. Scharlemann, and A. Thompson :
“Levelling an unknotting tunnel ,”
Geom. Topol.
4
(2000 ),
pp. 243–275 .
MR
1778174
Zbl
0958.57007
ArXiv
math/9910099
article
Abstract
People
BibTeX
It is a consequence of theorems of Gordon–Reid [1995] and Thompson [1997] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [1992], who showed that the (now known) classification of unknotting tunnels for 2-bridge knots would follow quickly if it were known that any unknotting tunnel can be made level.
@article {key1778174m,
AUTHOR = {Goda, Hiroshi and Scharlemann, Martin
and Thompson, Abigail},
TITLE = {Levelling an unknotting tunnel},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry and Topology},
VOLUME = {4},
YEAR = {2000},
PAGES = {243--275},
DOI = {10.2140/gt.2000.4.243},
NOTE = {ArXiv:math/9910099. MR:1778174. Zbl:0958.57007.},
ISSN = {1465-3060},
}
M. Scharlemann and A. Thompson :
“Unknotting tunnels and Seifert surfaces ,”
Proc. London Math. Soc. (3)
87 : 2
(2003 ),
pp. 523–544 .
MR
1990938
Zbl
1047.57008
ArXiv
math/0010212
article
Abstract
People
BibTeX
Let \( K \) be a knot with an unknotting tunnel \( \gamma \) and suppose that \( K \) is not a 2-bridge knot. There is an invariant
\[ \rho = p/q\in\mathbb{Q}/2\mathbb{Z} ,\]
with \( p \) odd, defined for the pair \( (K,\gamma) \) .
The invariant \( \rho \) has interesting geometric properties. It is often straightforward to calculate; for example, for \( K \) a torus knot and \( \gamma \) an annulus-spanning arc,
\[ \rho(K,\gamma) = 1 .\]
Although \( \rho \) is defined abstractly, it is naturally revealed when \( K\cup\gamma \) is put in thin position. If \( \rho\neq 1 \) then there is a minimal-genus Seifert surface \( F \) for \( K \) such that the tunnel \( \gamma \) can be slid and isotoped to lie on \( F \) . One consequence is that if
\[ \rho(K,\gamma)\neq 1 \]
then \( K > 1 \) . This confirms a conjecture of Goda and Teragaito for pairs \( (K,\gamma) \) with
\[ \rho(K,\gamma)\neq 1 .\]
@article {key1990938m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Unknotting tunnels and {S}eifert surfaces},
JOURNAL = {Proc. London Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {87},
NUMBER = {2},
YEAR = {2003},
PAGES = {523--544},
DOI = {10.1112/S0024611503014242},
NOTE = {ArXiv:math/0010212. MR:1990938. Zbl:1047.57008.},
ISSN = {0024-6115},
}
M. Scharlemann and A. Thompson :
“Thinning genus two Heegaard spines in \( S^3 \) ,”
J. Knot Theor. Ramif.
12 : 5
(2003 ),
pp. 683–708 .
MR
1999638
Zbl
1048.57002
article
Abstract
People
BibTeX
@article {key1999638m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Thinning genus two {H}eegaard spines
in \$S^3\$},
JOURNAL = {J. Knot Theor. Ramif.},
FJOURNAL = {Journal of Knot Theory and its Ramifications},
VOLUME = {12},
NUMBER = {5},
YEAR = {2003},
PAGES = {683--708},
DOI = {10.1142/S0218216503002706},
NOTE = {MR:1999638. Zbl:1048.57002.},
ISSN = {0218-2165},
}
M. Scharlemann and A. Thompson :
“On the additivity of knot width ,”
pp. 135–144
in
Proceedings of the Casson Fest
(Fayetteville, AR, 10–12 April 2003 and Austin, TX, 19–21 May 2003 ).
Edited by C. Gordon and Y. Rieck .
Geometry & Topology Monographs 7 .
Geometry & Topology Publications (Coventry, UK ),
2004 .
Based on the 28th University of Arkansas spring lecture series in the mathematical sciences and a conference on the topology of manifolds of dimensions 3 and 4. This paper was “Dedicated to Andrew Casson, a mathematician’s mathematician.”.
MR
2172481
Zbl
1207.57016
ArXiv
math/0403326
incollection
Abstract
People
BibTeX
It has been conjectured that the geometric invariant of knots in 3-space called the width is nearly additive. That is, letting \( w(K)\in 2\mathbb{N} \) denote the width of a knot \( K\subset S^3 \) , the conjecture is that
\[ w(K\#K^{\prime})=w(K)+w(K^{\prime})-2 .\]
We give an example of a knot \( K_1 \) so that for \( K_2 \) any 2-bridge knot, it appears that
\[ w(K_1\#K_2)=w(K_1) ,\]
contradicting the conjecture.
@incollection {key2172481m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {On the additivity of knot width},
BOOKTITLE = {Proceedings of the {C}asson {F}est},
EDITOR = {Gordon, Cameron and Rieck, Yoav},
SERIES = {Geometry \& Topology Monographs},
NUMBER = {7},
PUBLISHER = {Geometry \& Topology Publications},
ADDRESS = {Coventry, UK},
YEAR = {2004},
PAGES = {135--144},
DOI = {10.2140/gtm.2004.7.135},
NOTE = {(Fayetteville, AR, 10--12 April 2003
and Austin, TX, 19--21 May 2003). Based
on the 28th University of Arkansas spring
lecture series in the mathematical sciences
and a conference on the topology of
manifolds of dimensions 3 and 4. This
paper was ``Dedicated to Andrew Casson,
a mathematician's mathematician''. ArXiv:math/0403326.
MR:2172481. Zbl:1207.57016.},
ISSN = {1464-8989},
}
M. Scharlemann and A. Thompson :
“Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds ,”
Proc. Am. Math. Soc.
133 : 6
(2005 ),
pp. 1573–1580 .
MR
2120271
Zbl
1071.57015
ArXiv
math/0308011
article
Abstract
People
BibTeX
Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox’s theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the \( \partial \) -connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold’s complement.
@article {key2120271m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Surfaces, submanifolds, and aligned
{F}ox reimbedding in non-{H}aken 3-manifolds},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {133},
NUMBER = {6},
YEAR = {2005},
PAGES = {1573--1580},
DOI = {10.1090/S0002-9939-04-07704-4},
NOTE = {ArXiv:math/0308011. MR:2120271. Zbl:1071.57015.},
ISSN = {0002-9939},
}
M. Scharlemann and A. A. Thompson :
“Surgery on a knot in (surface \( \times I \) ) ,”
Algebr. Geom. Topol.
9 : 3
(2009 ),
pp. 1825–1835 .
MR
2550096
Zbl
1197.57011
ArXiv
0807.0405
article
Abstract
People
BibTeX
Suppose \( F \) is a compact orientable surface, \( K \) is a knot in \( F\times I \) , and \( (F\times I)_{\textrm{surg}} \) is the 3-manifold obtained by some nontrivial surgery on \( K \) . If \( F\times\{0\} \) compresses in \( (F\times I)_{\textrm{surg}} \) , then there is an annulus in \( F\times I \) with one end \( K \) and the other end an essential simple closed curve in \( F\times\{0\} \) . Moreover, the end of the annulus at \( K \) determines the surgery slope.
An application: Suppose \( M \) is a compact orientable 3-manifold that fibers over the circle. If surgery on \( K\subset M \) yields a reducible manifold, then either
the projection \( K\subset M\to S^1 \) has nontrivial winding number,
\( K \) lies in a ball,
\( K \) lies in a fiber, or
\( K \) is cabled.
@article {key2550096m,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail
A.},
TITLE = {Surgery on a knot in (surface \$\times
I\$)},
JOURNAL = {Algebr. Geom. Topol.},
FJOURNAL = {Algebraic \& Geometric Topology},
VOLUME = {9},
NUMBER = {3},
YEAR = {2009},
PAGES = {1825--1835},
DOI = {10.2140/agt.2009.9.1825},
NOTE = {ArXiv:0807.0405. MR:2550096. Zbl:1197.57011.},
ISSN = {1472-2747},
}
M. Scharlemann and A. Thompson :
Fibered knots and Property 2R .
Preprint ,
2009 .
ArXiv
0901.2319
techreport
Abstract
People
BibTeX
It is shown, using sutured manifold theory, that if there are any 2-component counterexamples to the Generalized Property R Conjecture, then any knot of least genus among components of such counterexamples is not a fibered knot. The general question of what fibered knots might appear as a component of such a counterexample is further considered; much can be said about the monodromy of the fiber, particularly in the case in which the fiber is of genus two.
@techreport {key0901.2319a,
AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
TITLE = {Fibered knots and {P}roperty 2R},
TYPE = {preprint},
YEAR = {2009},
PAGES = {18},
NOTE = {ArXiv:0901.2319.},
}
R. E. Gompf, M. Scharlemann, and A. Thompson :
“Fibered knots and potential counterexamples to the property \( 2{R} \) and slice-ribbon conjectures ,”
Geom. Topol.
14 : 4
(2010 ),
pp. 2305–2347 .
MR
2740649
Zbl
1214.57008
ArXiv
1103.1601
article
Abstract
People
BibTeX
If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions.
The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to \( \#_2(S^1\times S^2) \) . We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.
@article {key2740649m,
AUTHOR = {Gompf, Robert E. and Scharlemann, Martin
and Thompson, Abigail},
TITLE = {Fibered knots and potential counterexamples
to the property \$2{R}\$ and slice-ribbon
conjectures},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry \& Topology},
VOLUME = {14},
NUMBER = {4},
YEAR = {2010},
PAGES = {2305--2347},
DOI = {10.2140/gt.2010.14.2305},
NOTE = {ArXiv:1103.1601. MR:2740649. Zbl:1214.57008.},
ISSN = {1465-3060},
}
J. H. Rubinstein and A. Thompson :
“3-manifolds with Heegaard splittings of distance two ,”
pp. 341–346
in
Geometry and topology down under .
Edited by C. D. Hodgson, W. H. Jaco, M. G. Scharlemann, and S. Tillman .
Contemporary Mathematics 597 .
American Mathematial Society (Providence, RI ),
2013 .
MR
3186682
incollection
Abstract
People
BibTeX
It is well-known that the distance of a Heegard splitting, as defined by Hempel [2001], is related to global properties of the underlying 3-manifold. For example, Casson–Gordon [1987] show that existence of a splitting of distance one implies that the manifold is Haken, i.e. has an embedded incompressible surface. Distance zero splittings are either stabilized, or the underlying manifold is reducible. In this paper, we study the case of distance two splittings. These are interesting as many known examples of splittings of distance greater than one are actually of distance two. Our aim is to describe how simple conditions on a distance two splitting imply interesting properties of the underlying manifold.
@incollection {key3186682m,
AUTHOR = {Rubinstein, J. Hyam and Thompson, Abigail},
TITLE = {3-manifolds with {H}eegaard splittings
of distance two},
BOOKTITLE = {Geometry and topology down under},
EDITOR = {Hodgson, Craig D. and Jaco, William
H. and Scharlemann, Martin G. and Tillman,
Stephan},
SERIES = {Contemporary Mathematics},
NUMBER = {597},
PUBLISHER = {American Mathematial Society},
ADDRESS = {Providence, RI},
YEAR = {2013},
PAGES = {341--346},
DOI = {10.1090/conm/597/11878},
NOTE = {MR:3186682.},
ISSN = {0271-4132},
ISBN = {9780821884805},
}