Ulrich Oertel


Fall, 2018
Office - Smith Hall 322

Courses-

    21:640:350:01  Linear Algebra,  Monday, Wednesday 10:00-11:20 pm,  Room 244,  Smith Hall

    21:640:441:01 Topology I, Monday 2:30-3:50, Wednesday 1:00-2:20, Room 240, Smith Hall

Course web pages on Blackboard

Office Hours:  Monday,4:15- 5:15 pm, Wednesday  5:30-6:30 pm and by appointment

Office Phone - 973-353-3909;           Email - oertel@newark.rutgers.edu

Department of Mathematics and Computer Science, Rutgers-Newark


Current Research 



Lamination knots and links

(This brief exposition is designed for a general audience.)

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 preferred framing.

trefoil

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 picture. 

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 arXiv

Measured lamination spaces for surface pairs.  Math arXiv

Finite height lamination spaces for surfaces.  Math arXiv


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
 

Normal surfaces

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