M. Culler, W. Jaco, and H. Rubinstein :
“Incompressible surfaces in once-punctured torus bundles Proc. London Math. Soc. (3)
45 : 3
(1982 ),
pp. 385–419 .
MR
675414 Zbl
0515.57002 article
People
BibTeX
@article {key675414m,
AUTHOR = {Culler, M. and Jaco, W. and Rubinstein,
H.},
TITLE = {Incompressible surfaces in once-punctured
torus bundles},
JOURNAL = {Proc. London Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {45},
NUMBER = {3},
YEAR = {1982},
PAGES = {385--419},
DOI = {10.1112/plms/s3-45.3.385},
NOTE = {MR:675414. Zbl:0515.57002.},
ISSN = {0024-6115},
CODEN = {PLMTAL},
}
W. Jaco and J. H. Rubinstein :
“A piecewise linear theory of minimal surfaces in 3-manifolds ,”
pp. 99–110
in
Miniconference on geometry and partial differential equations
(Canberra, 1–3 August 1985 ).
Edited by L. M. Simon and N. S. Trudinger .
Proceedings of the Centre for Mathematical Analysis, Australian National University 10 .
Australian National University (Canberra ),
1986 .
First miniconference.
MR
857657 Zbl
0596.53007 incollection
People
BibTeX
@incollection {key857657m,
AUTHOR = {Jaco, William and Rubinstein, J. H.},
TITLE = {A piecewise linear theory of minimal
surfaces in 3-manifolds},
BOOKTITLE = {Miniconference on geometry and partial
differential equations},
EDITOR = {Simon, Leon M. and Trudinger, Neil S.},
SERIES = {Proceedings of the Centre for Mathematical
Analysis, Australian National University},
NUMBER = {10},
PUBLISHER = {Australian National University},
ADDRESS = {Canberra},
YEAR = {1986},
PAGES = {99--110},
NOTE = {(Canberra, 1--3 August 1985). First
miniconference. MR:857657. Zbl:0596.53007.},
ISBN = {9780867845112},
}
W. Jaco and J. H. Rubinstein :
“PL minimal surfaces in 3-manifolds J. Differ. Geom.
27 : 3
(1988 ),
pp. 493–524 .
MR
940116 Zbl
0652.57005 article
People
BibTeX
@article {key940116m,
AUTHOR = {Jaco, William and Rubinstein, J. Hyam},
TITLE = {P{L} minimal surfaces in 3-manifolds},
JOURNAL = {J. Differ. Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {27},
NUMBER = {3},
YEAR = {1988},
PAGES = {493--524},
URL = {http://projecteuclid.org/euclid.jdg/1214442006},
NOTE = {MR:940116. Zbl:0652.57005.},
ISSN = {0022-040X},
CODEN = {JDGEAS},
}
W. Jaco and J. H. Rubinstein :
“PL equivariant surgery and invariant decompositions of 3-manifolds Adv. in Math.
73 : 2
(1989 ),
pp. 149–191 .
MR
987273 Zbl
0682.57005 article
People
BibTeX
@article {key987273m,
AUTHOR = {Jaco, William and Rubinstein, J. Hyam},
TITLE = {P{L} equivariant surgery and invariant
decompositions of 3-manifolds},
JOURNAL = {Adv. in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {73},
NUMBER = {2},
YEAR = {1989},
PAGES = {149--191},
DOI = {10.1016/0001-8708(89)90067-4},
NOTE = {MR:987273. Zbl:0682.57005.},
ISSN = {0001-8708},
CODEN = {ADMTA4},
}
W. Jaco, D. Letscher, and J. H. Rubinstein :
“Algorithms for essential surfaces in 3-manifolds 107–124
in
Topology and geometry: Commemorating SISTAG
(National University of Singapore, 2–6 July 2001 ).
Edited by A. J. Berrick, M. C. Leung, and X. Xu .
Contemporary Mathematics 314 .
American Mathematical Society (Providence, RI ),
2002 .
Singapore International Symposium in Topology and Geometry.
MR
1941626 Zbl
1012.57029 incollection
Abstract
People
BibTeX
In this paper we outline several algorithms to find essential surfaces in 3-dimensional manifolds. In particular, the classical decomposition theorems of 3-manifolds (Kneser–Milnor connected sum decomposition and the JSJ decomposition) are defined by splitting along families of disjoint essential spheres and tori. We give algorithms to find such surfaces, using normal and almost normal surface theory and the technique of crushing triangulations. These algorithms have running time \( O(p(t)3^t) \) , where \( t \) is the number of tetrahedra in any given initial one-vertex triangulation of the manifold and \( p(t) \) is some polynomial in \( t \) . A special instance of these ideas gives a new algorithm also with running time \( O(p(t)3^t) \) for deciding if a knot is the unknot, where \( t \) is the number of tetrahedra
in an ideal triangulation of the knot complement. Note that there is a bound \( t \leq cn \) , where \( n \) is the crossing number of a projection of the knot and \( c \) is a (small) constant. We discuss this in detail elsewhere. Note that these algorithms avoid the computationally more expensive issue of deciding whether a given surface is incompressible.
Our other main algorithm is to determine if a given 3-manifold has an embedded incompressible surface or not. If the manifold is known to be irreducible (by applying our first algorithm), then this is the same as determining if it is Haken or not. As Thurston’s uniformisation theorem applies to the class of Haken 3-manifolds, this is a key algorithmic issue in 3-manifold theory. In particular, few examples are known of non-Haken 3-manifolds and we hope that this algorithm will be useful for finding new ones.
This algorithm has running time \( O(k^t) \) , where \( k \) is a constant. We will give a rough upper bound on \( k \) and in another paper discuss some lower bounds for various important quantities involved in normal and almost normal surface theory.
A. Casson gave inspirational lectures at Montreal in 1995 and at the Technion in 1999 on related topics. In particular he outlined an approach to the problem of finding the connected sum decomposition in the latter talk and introduced linear programming as a key tool. He also described crushing normal surfaces in the former talk, as a way of simplifying triangulations. We will discuss his method and compare it to ours.
@incollection {key1941626m,
AUTHOR = {Jaco, William and Letscher, David and
Rubinstein, J. Hyam},
TITLE = {Algorithms for essential surfaces in
3-manifolds},
BOOKTITLE = {Topology and geometry: {C}ommemorating
{SISTAG}},
EDITOR = {Berrick, A. J. and Leung, Man Chun and
Xu, Xingwang},
SERIES = {Contemporary Mathematics},
NUMBER = {314},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2002},
PAGES = {107--124},
DOI = {10.1090/conm/314/05426},
NOTE = {(National University of Singapore, 2--6
July 2001). Singapore International
Symposium in Topology and Geometry.
MR:1941626. Zbl:1012.57029.},
ISSN = {0271-4132},
ISBN = {9780821856505},
}
W. Jaco and J. H. Rubinstein :
“0-efficient triangulations of 3-manifolds J. Differ. Geom.
65 : 1
(2003 ),
pp. 61–168 .
MR
2057531 Zbl
1068.57023 ArXiv
0207158 article
Abstract
People
BibTeX
0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold \( M \) can be modified to a 0-efficient triangulation or \( M \) can be shown to be one of the manifolds \( \mathbb{S}^3 \) , \( \mathbb{R}P^3 \) or \( L(3,1) \) . Similarly, any triangulation of a compact, orientable, irreducible, \( \partial \) -irreducible 3-manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal triangulation is defined. It is shown if \( M \) is a compact, orientable, irreducible, \( \partial \) -irreducible 3-manifold having no essential annuli and distinct from the 3-cell, then \( \mathring{M} \) admits an ideal triangulation; furthermore, it is shown that any ideal triangulation of such a 3-manifold can be modified to a 0-efficient ideal triangulation. A 0-efficient triangulation of a closed manifold has only one vertex or the manifold is \( \mathbb{S}^3 \) and the triangulation has precisely two vertices. 0-efficient triangulations of 3-manifolds with boundary, and distinct from the 3-cell, have all their vertices in the boundary and then just one vertex in each boundary component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal surface. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-manifold, an alternate algorithm for the 3-sphere recognition problem, results on edges of low valence in minimal triangulations of 3-manifolds, and a construction of irreducible knots in closed 3-manifolds.
@article {key2057531m,
AUTHOR = {Jaco, William and Rubinstein, J. Hyam},
TITLE = {0-efficient triangulations of 3-manifolds},
JOURNAL = {J. Differ. Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {65},
NUMBER = {1},
YEAR = {2003},
PAGES = {61--168},
URL = {http://projecteuclid.org/euclid.jdg/1090503053},
NOTE = {ArXiv:0207158. MR:2057531. Zbl:1068.57023.},
ISSN = {0022-040X},
CODEN = {JDGEAS},
}
W. Jaco and J. H. Rubinstein :
Layered-triangulations of 3-manifolds .
Preprint ,
March 2006 .
ArXiv
0603601 techreport
Abstract
People
BibTeX
A family of one-vertex triangulations of the genus-\( g \) -handlebody, called layered-triangulations, is defined. These triangulations induce a one-vertex triangulation on the boundary of the handlebody, a genus \( g \) surface. Conversely, any one-vertex triangulation of a genus \( g \) surface can be placed on the boundary of the genus-\( g \) -handlebody in infinitely many distinct ways; it is shown that any of these can be extended to a layered-triangulation of the handlebody. To organize this study, a graph is constructed, for each genus \( g\geq 1 \) , called the \( L_g \) graph; its 0-cells are in one-one correspondence with equivalence classes (up to homeomorphism of the handlebody) of one-vertex triangulations of the genus \( g \) surface on the boundary of the handlebody and its 1-cells correspond to the operation of a diagonal flip (or \( 2\leftrightarrow 2 \) Pachner move) on a one-vertex triangulation of a surface. A complete and detailed
analysis of layered-triangulations is given in the case of the solid torus (\( g = 1 \) ), including the classification of all normal and almost normal surfaces in these triangulations. An initial investigation of normal and almost normal surfaces in layered-triangulations of higher genera handlebodies is discussed. Using Heegaard splittings, layered-triangulations of handlebodies can be used to construct special one-vertex triangulations of 3-manifolds, also called layered-triangulations. Minimal layered-triangulations of lens spaces (genus one manifolds) provide a common setting for new proofs of the classification of lens spaces admitting an embedded non orientable surface and the classification of embedded non orientable surfaces in each such lens space, as well as a new proof of the uniqueness of Heegaard splittings of lens spaces, including \( \mathbb{S}^3 \) and \( \mathbb{S}^2\times \mathbb{S}^1 \) . Canonical triangulations of Dehn fillings called triangulated Dehn fillings are constructed and applied to the study of Heegaard splittings and efficient triangulations of Dehn fillings. It is shown that all closed 3-manifolds can be presented in a new way, and with very nice triangulations, using layered-triangulations of handlebodies that have special one-vertex triangulations of a closed surface on their boundaries, called 2-symmetric triangulations. We provide a quick introduction to a connection between layered-triangulations and foliations. Numerous questions remain unanswered, particularly in relation to the \( L_g \) -graph, 2-symmetric triangulations of a closed orientable surface, minimal layered-triangulations of genus-\( g \) -handlebodies, \( g\geq 1 \) and the relationship of layered-triangulations to foliations.
@techreport {key0603601a,
AUTHOR = {Jaco, W. and Rubinstein, J. H.},
TITLE = {Layered-triangulations of 3-manifolds},
TYPE = {Preprint},
MONTH = {March},
YEAR = {2006},
PAGES = {97},
NOTE = {ArXiv:0603601.},
}
W. Jaco, J. H. Rubinstein, and S. Tillmann :
\( \mathbb{Z}_2 \) -Thurston norm and complexity of 3-manifoldsPreprint ,
June 2009 .
ArXiv
0906.4864 techreport
Abstract
People
BibTeX
@techreport {key0906.4864a,
AUTHOR = {Jaco, William and Rubinstein, J. Hyam
and Tillmann, Stephan},
TITLE = {\$\mathbb{Z}_2\$-{T}hurston norm and complexity
of 3-manifolds},
TYPE = {Preprint},
MONTH = {June},
YEAR = {2009},
PAGES = {19},
NOTE = {ArXiv:0906.4864.},
}
W. Jaco, H. Rubinstein, and S. Tillmann :
“Minimal triangulations for an infinite family of lens spaces J. Topol.
2 : 1
(2009 ),
pp. 157–180 .
MR
2499441 Zbl
1227.57026 ArXiv
0805.2425 article
Abstract
People
BibTeX
The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein, and unless the lens space is \( L(3,1) \) , a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its minimal layered triangulation . This paper proves that for each \( n \geq 2 \) , the minimal layered triangulation of the lens space \( L(2n,1) \) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens space of the form \( L(2n,1) \) .
@article {key2499441m,
AUTHOR = {Jaco, William and Rubinstein, Hyam and
Tillmann, Stephan},
TITLE = {Minimal triangulations for an infinite
family of lens spaces},
JOURNAL = {J. Topol.},
FJOURNAL = {Journal of Topology},
VOLUME = {2},
NUMBER = {1},
YEAR = {2009},
PAGES = {157--180},
DOI = {10.1112/jtopol/jtp004},
NOTE = {ArXiv:0805.2425. MR:2499441. Zbl:1227.57026.},
ISSN = {1753-8416},
}
W. Jaco, J. H. Rubinstein, and E. Sedgwick :
“Finding planar surfaces in knot- and link-manifolds J. Knot Theory Ramifications
18 : 3
(2009 ),
pp. 397–446 .
MR
2514851 Zbl
1176.57024 ArXiv
0608700 article
Abstract
People
BibTeX
It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. Two similar results are obtained with added boundary conditions. Namely, given a link-manifold \( M \) , a component \( B \) of \( \partial M \) , and a slope \( \gamma \) on \( B \) , there is an algorithm to decide if there is an embedded punctured-disk in \( M \) with boundary \( \gamma \) and punctures in \( \partial M\backslash B \) ; and given a link-manifold \( M \) , a component \( B \) of \( \partial M \) , and a meridian slope \( \mu \) on \( B \) , there is an algorithm to decide if there is an embedded punctured-disk with boundary a longitude on \( B \) and punctures in \( \partial M\backslash B \) . In both cases, if there is one, the algorithm will construct one. The proofs introduce a number of new methods and differ from the classical proofs, using normal surfaces, as solutions may not be found among the fundamental solutions.
@article {key2514851m,
AUTHOR = {Jaco, William and Rubinstein, J. Hyam
and Sedgwick, Eric},
TITLE = {Finding planar surfaces in knot- and
link-manifolds},
JOURNAL = {J. Knot Theory Ramifications},
FJOURNAL = {Journal of Knot Theory and its Ramifications},
VOLUME = {18},
NUMBER = {3},
YEAR = {2009},
PAGES = {397--446},
DOI = {10.1142/S0218216509006987},
NOTE = {ArXiv:0608700. MR:2514851. Zbl:1176.57024.},
ISSN = {0218-2165},
}
J. H. Rubinstein :
“Problems at the Jacofest 195–196
in
Topology and geometry in dimension three: Triangulations, invariants, and geometric structures
(Oklahoma State University, Stillwater, OK, 4–6 June 2010 ).
Edited by W. Li, L. Bartolini, J. Johnson, F. Luo, R. Myers, and J. H. Rubinstein .
Contemporary Mathematics 560 .
American Mathematical Society (Providence, RI ),
2011 .
Conference in honor of William Jaco’s 70th birthday.
MR
2866932 incollection
People
BibTeX
@incollection {key2866932m,
AUTHOR = {Rubinstein, J. Hyam},
TITLE = {Problems at the {J}acofest},
BOOKTITLE = {Topology and geometry in dimension three:
{T}riangulations, invariants, and geometric
structures},
EDITOR = {Li, Weiping and Bartolini, Loretta and
Johnson, Jesse and Luo, Feng and Myers,
Robert and Rubinstein, J. Hyam},
SERIES = {Contemporary Mathematics},
NUMBER = {560},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2011},
PAGES = {195--196},
DOI = {10.1090/conm/560/11100},
NOTE = {(Oklahoma State University, Stillwater,
OK, 4--6 June 2010). Conference in honor
of {W}illiam {J}aco's 70th birthday.
MR:2866932.},
ISSN = {0271-4132},
ISBN = {9780821852958},
}
Topology and geometry in dimension three: Triangulations, invariants, and geometric structures Oklahoma State University, Stillwater, OK, 4–6 June 2010 ).
Edited by W. Li, L. Bartolini, J. Johnson, F. Luo, R. Myers, and J. H. Rubinstein .
Contemporary Mathematics 560 .
American Mathematical Society (Providence, RI ),
2011 .
Conference in honor of William Jaco’s 70th birthday.
MR
2866933 Zbl
1231.57001 book
People
BibTeX
@book {key2866933m,
TITLE = {Topology and geometry in dimension three:
{T}riangulations, invariants, and geometric
structures},
EDITOR = {Li, Weiping and Bartolini, Loretta and
Johnson, Jesse and Luo, Feng and Myers,
Robert and Rubinstein, J. Hyam},
SERIES = {Contemporary Mathematics},
NUMBER = {560},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2011},
PAGES = {x+196},
DOI = {10.1090/conm/560},
NOTE = {(Oklahoma State University, Stillwater,
OK, 4--6 June 2010). Conference in honor
of {W}illiam {J}aco's 70th birthday.
MR:2866933. Zbl:1231.57001.},
ISSN = {0271-4132},
ISBN = {9780821852958},
}
W. Jaco and J. H. Rubinstein :
Annular-efficient triangulations of 3-manifolds .
Preprint ,
August 2011 .
ArXiv
1108.2936 techreport
Abstract
People
BibTeX
A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, irreducible, boundary-irreducible, and an-annular 3-manifold, any triangulation can be modified to an annular-efficient triangulation. It follows that for a manifold satisfying this hypothesis, there are only a finite number of boundary slopes for incompressible and boundary-incompressible surfaces of a bounded Euler characteristic.
@techreport {key1108.2936a,
AUTHOR = {Jaco, William and Rubinstein, J. Hyam},
TITLE = {Annular-efficient triangulations of
3-manifolds},
TYPE = {Preprint},
MONTH = {August},
YEAR = {2011},
PAGES = {21},
NOTE = {ArXiv:1108.2936.},
}
W. Jaco, J. H. Rubinstein, and S. Tillmann :
“Coverings and minimal triangulations of 3-manifolds Algebr. Geom. Topol.
11 : 3
(2011 ),
pp. 1257–1265 .
MR
2801418 Zbl
1229.57010 ArXiv
0903.0112 article
Abstract
People
BibTeX
This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space \( L(4k,2k-1) \) and the generalised quaternionic space \( \mathbb{S}^3/Q_{4k} \) have complexity \( k \) , where \( k \geq 2 \) . Moreover, it is shown that their minimal triangulations are unique.
@article {key2801418m,
AUTHOR = {Jaco, William and Rubinstein, J. Hyam
and Tillmann, Stephan},
TITLE = {Coverings and minimal triangulations
of 3-manifolds},
JOURNAL = {Algebr. Geom. Topol.},
FJOURNAL = {Algebraic \& Geometric Topology},
VOLUME = {11},
NUMBER = {3},
YEAR = {2011},
PAGES = {1257--1265},
DOI = {10.2140/agt.2011.11.1257},
NOTE = {ArXiv:0903.0112. MR:2801418. Zbl:1229.57010.},
ISSN = {1472-2747},
}
B. Foozwell and H. Rubinstein :
“Introduction to the theory of Haken \( n \) -manifolds 71–84
in
Topology and geometry in dimension three: Triangulations, invariants, and geometric structures
(Oklahoma State University, Stillwater, OK, 4–6 June 2010 ).
Edited by W. Li, L. Bartolini, J. Johnson, F. Luo, R. Myers, and J. H. Rubinstein .
Contemporary Mathematics 560 .
American Mathematical Society (Providence, RI ),
2011 .
Conference in honor of William Jaco’s 70th birthday.
MR
2866924 incollection
Abstract
People
BibTeX
We define the class of Haken \( n \) -manifolds, following Johannson [1994]. A number of basic results are proved and some examples given. A key property is that these manifolds have universal coverings \( \mathbf{R}^n \) and so are aspherical. The latter is established here and the former is proved in [Foozwell 2007]. Some problems are given in the final section. In particular, there is a natural Haken cobordism category and computing this would provide many interesting examples.
@incollection {key2866924m,
AUTHOR = {Foozwell, Bell and Rubinstein, Hyam},
TITLE = {Introduction to the theory of {H}aken
\$n\$-manifolds},
BOOKTITLE = {Topology and geometry in dimension three:
{T}riangulations, invariants, and geometric
structures},
EDITOR = {Li, Weiping and Bartolini, Loretta and
Johnson, Jesse and Luo, Feng and Myers,
Robert and Rubinstein, J. Hyam},
SERIES = {Contemporary Mathematics},
NUMBER = {560},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2011},
PAGES = {71--84},
DOI = {10.1090/conm/560/11092},
NOTE = {(Oklahoma State University, Stillwater,
OK, 4--6 June 2010). Conference in honor
of {W}illiam {J}aco's 70th birthday.
MR:2866924.},
ISSN = {0271-4132},
ISBN = {9780821852958},
}
W. H. Jaco and J. H. Rubinstein :
Inflations of ideal triangulations .
Preprint ,
February 2013 .
ArXiv
1302.6921 techreport
Abstract
People
BibTeX
Starting with an ideal triangulation of the interior of a compact 3-manifold \( M \) with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulations of \( M \) itself. Besides a step-by-step algorithm for such a construction, we provide examples of an inflation of the two-tetrahedra ideal triangulation of the complement of the figure-eight knot in the 3-sphere, giving a minimal triangulation, having ten tetrahedra, of the figure-eight knot exterior. As another example, we provide an inflation of the one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven tetrahedra, of a nonorientable compact 3-manifold with Klein bottle boundary. Several applications of inflations are discussed.
@techreport {key1302.6921a,
AUTHOR = {Jaco, William H. and Rubinstein, J.
Hyam},
TITLE = {Inflations of ideal triangulations},
TYPE = {Preprint},
MONTH = {February},
YEAR = {2013},
PAGES = {48},
NOTE = {ArXiv:1302.6921.},
}
