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21:640:441:01 Topology I, Monday 2:30-3:50,
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of Mathematics and Computer Science, Rutgers-Newark
Lamination knots and links
(This brief exposition is designed for a
Below in (a) we show a right handed trefoil
knot K in 3-space or the
3-sphere. There is a simple algorithm, due to Herbert
Seifert for finding a Seifert surface.
A Seifert surface for an (oriented) knot or link is an oriented surface
whose boundary is the knot or link. Perhaps the picture is
not completely clear. The surface is constructed from two
truncated triangles. We fold the truncated vertices of the
lower triangle (lightly stippled on top) over. In (b)
you see the folded-over corners as boldly stippled regions.
The second smaller truncated triangle floats above the first one and is
lightly stippled. Finally, we perform a quarter-twist on each
truncated corner of each triangle and glue the edges where the
triangles were truncated. The result is an orientable surface
with one side boldly stippled, the other side lightly
stippled. This is a Seifert surface S for the knot K. Now we fatten the
Seifert surface S,
as shown in (c). The fattened Seifert surface looks
complicated in the figure, where it is shown embedded in 3-space, but
abstractly it is quite a simple space, a product of a surface with
boundary and an interval. Finally, in (d)
we show the edge of this fattened Seifert surface. This is
a framed knot, i.e. a knot
"made from a ribbon," where the ribbon has some twisting.
Obviously you can put more or less twisting in this ribbon, in either
sense, yielding different framed knots. It turns out there
is only one framing which can be the edge of any fattened Seifert
surface for a given knot. This is the unique
I am studying lamination knots and links.
An example is shown in red under the "Current Research" heading
above. Lamination links are always framed, or ribbon-like, but
now the ribbons are joined like a freeway in 3-space, twisting,
turning, and not always facing up. Also, the
ribbons have specified positive widths which "add up" correctly where
the ribbon branches, just as the widths of segments of freeway
(measured in numbers of lanes) add up correctly at freeway branchings
(if no lanes end). In our setting, widths do not need to be
positive integers; they could be positive real numbers. We've
assigned some widths in the figure which add up correctly at
branchings. The framed link has one more property: It is
oriented, so the traffic flows in the same direction in all lanes,
giving a one-way freeway. Actually, the lamination link is a more
abstract object which is represented by the object in the
So now it is natural to ask whether this lamination link is the boundary of something like a Seifert surface. As in the case of classical knots, most (framed or ribbon-like) lamination links do not bound a Seifert laminations, but our example does bound a Seifert lamination represented by the branched surface with weights shown below:
Notice that the weights again "add up correctly" at the branch locus of the branched surface. You can try to convince yourself that the "edge" of the fattened branched surface is the same as the original picture of the red lamination link above.
In my research, I have addressed some obvious questions concerning lamination links. In the paper "Lamination links in 3-manifolds" Math arXiv, I introduce lamination links. For a classical oriented link, we ask what is the "simplest" Seifert surface bounded by the link. "Simplicity" is measured by a number associated to the surface, which is called the genus. Thus we ask, "What is the minimal genus of a Seifert surface." Assuming a lamination link bounds a Seifert lamination, there is an associated number called the Euler characteristic, related to genus. I show that it is possible to find a Seifert lamination which is "simplest" as measured by this number. In the same paper, I also begin the work needed to construct a "space of lamination links," whose points represent lamination links.
In the paper "A Seifert algorithm for lamination links" Math arXiv, I consider the question whether a given lamination link bounds a Seifert lamination. In the classical case, a framed oriented link usually does not bound a Seifert surface, but if we allow a "change of framing" or "change of twisting," then we can guarantee the existence of a Seifert surface. The situation is similar for lamination links. Again, if we are given a framed lamination link, we can show that it bounds a Seifert lamination after we modify it suitably. In fact, we can construct the Seifert lamination for the modified link explicitly. This is a generalization of Seifert's algorithm.
Laminations with transverse measures in ordered abelian semigroups.
In an attempt to understand "finite height (or finite depth) measured laminations," I use transverse measures on the laminations with values in certain ordered abelian semi-groups and ordered semi-rings. I describe a wide variety of ordered algebraic structures which can be used to give transverse measures or structures to laminations. Even if a lamination does not admit one of these transverse measures, a lift to a covering space may admit such a measure. The result is that one can use these measures to describe large classes of laminations.
I began this work by trying to describe spaces of finite height
measured laminations in surfaces. The earlier papers now
need to be revised.
Laminations with transverse measures in ordered abelian semi-groups. Math arXivMeasured lamination spaces for surface pairs. Math arXiv
Finite height lamination spaces for
Automorphisms of 3-manifolds
The most interesting automorphisms (self-homeomorphisms) of 3-manifolds occur in boundary-reducible and reducible manifolds. The paper Automorphisms of 3-dimensional handlebodies develops a theory of automorphisms of handlebodies and compression bodies analogous to the Nielsen-Thurston theory of autmorphisms of surfaces. In the theory, the analogue of the pseudo-Anosov automorphism is the generic automorphism of a handlebody or compression body, and generic automorphisms turn out, as one would expect, to be the most interesting and mysterious. Automorphisms of irreducible 3-manifolds with non-empty boundary can then be understood by combining the new methods with older methods in 3-manifold topology, involving the Jaco-Shalen-Johannson characteristic submanifold and Bonahon's characteristic compression body. Much remains to be understood about generic automorphisms, which are currently also being studied by former Rutgers graduate student Leonardo Navarro de Carvalho.
In work with Carvalho, the ideas used for the classification of automorphisms of handlebodies and compression bodies are extended to classify, in a certain sense, the automorphisms of any compact 3-manifold (satisfying the Thurston Geometrization Conjecture). The current form of the paper is not satisfactory, and we plan to revise it extensively before publishing.
Automorphisms of 3-dimensional handlebodies and compression bodies, Topology 41 (2002) 363-410: Postscript file PDF file
A classification of automorphisms of
compact 3-manifolds, with Leonardo Navarro de
Carvalho. Math arXiv
Mapping class groups of compression bodies and 3-manifolds
In work related to the classification of automorphisms of 3-manifolds, I obtained results concerning the mapping class associated to a compression bodies. This is applied to obtain results concerning the mapping class group of a reducible 3-manifold M. The posted manuscript has been changed extensively, and the goals for the final version have become more and more ambitious. This paper is helpful for making a classification of automorphisms (as described above) more natural and appealing.
Mapping class groups of compression bodies and 3-manifolds. Math arXiv
Charalampos Charitos and I have a new normal surface theory for
incompressible (but not necessarily boundary incompressible) surfaces
in irrecucible (but not necessarily boundary irreducible) compact
3-manifolds with boundary. It may be possible to apply this
theory to obtain better invariant lamination-like objects for generic
automorphisms of handlebodies and compression bodies. One of the
motivations for the study of finite height measured laminations (above)
is that these invariant lamination-like objects should be finite height
measured in the appropriate sense.
Essential disks and semi-essential surfaces in 3-manifolds, with Charalampos Charitos. Essential disks and semi-essential surfaces in 3-manifolds, Topology Appl. 159 (2012), no. 8, 2174-2186. Math arXiv
Contact structures and contaminations in 3-manifolds
In a project with Jacek Swiatkowski (Wroclaw University) I am studying contact structures using branched surface techniques. A natural generalization of the contact structure and the confoliation is the contamination, an object which can be carried by a branched surface. Contact structures can be represented by contaminations. The goals of this research are 1) to understand contact structures combinatorially; 2) to understand the relationships between foliations, confoliations, contact structures, laminations, and contaminations.
A contamination carrying criterion for branched surfaces, with J. Swiatkowski, Ann. Global Anal. Geom. 34 (2008), no. 2, 135-152. Math arXiv
Correction: "A contamination carrying criterion for branched surfaces," with J. Swiatkowski, Ann. Global Anal. Geom. 44 (2013), no. 2, 137-150.
Contact structures, sigma-confoliations, and contaminations in
3-manifolds, with J. Swiatkowski, Commun. Contemp. Math. 11
(2009), no. 2, 201-264.. Math arXiv
Some Other Papers