# Mark Brittenham

Professor Mathematics mbrittenham2@unl.edu 329 Avery Hall

(very old!) CV

Class Websites

## Research

Below are some of my research papers and preprints:

• Tame filling invariants for groups (with S. Hermiller), submitted. PDF
• A uniform model for almost convexity and rewriting systems (with S. Hermiller), submitted. PDF
• Algorithms and topology for Cayley graphs of groups (with S. Hermiller and D. Holt) to appear in J. Algebra. PDF
• 4-moves and the Dabkowski-Sahi invariant for knots (with S. Hermiller and R. Todd). J. Knot Theory Ramifications 22 (2013) 1350069.1–20. PDF
• Subgroups of free idempotent generated semigroups: full linear monoid (with Stuart Margolis and John Meakin). arXiv:1009.5683. PDF
• Subgroups of free idempotent generated semigroups need not be free (with S. Margolis and J. Meakin). J. Algebra 321 (2009) 3026–3042. PDF
• Knots with unique minimal genus Seifert surface and depth of knots. J. Knot Thy. Ram. 17 (2008) 315–335. In this paper we use the construction of knots with genus one free Seifert surfaces (again) to create families of hyperbolic knots which each have a unique minimal genus Seifert surface which cannot be the sole compact leaf of a depth one foliation. PDF (with black and white figures - more printable) PDF with color figures
• Families of knots for which Morton's inequality is strict (with J. Jensen). Comm. Anal. Geom. 15 (2007) 971–983. We show how to build infinite families of knots, each having the maximum degree of its HOMFLY polynomial strictly less than twice its canonical genus (i.e., Morton's inequality is strict). The families are based on a small handful of examples discovered by Stoimenow.PDF
• The Heegaard genus of bundles over S^1 (with Yo'av Rieck). Geometry and Topology Monographs 12 (2007) 17–35. PDF
• Canonical genus and the Whitehead doubles of families of alternating knots (with Jacqueline Jensen). math.GT/0608765. PDF
• Essential laminations and branched surfaces in the exteriors of links (with C. Hayashi, M. Hirasawa, T. Kobayashi, and K. Shimokawa). Japanese Journal of Mathematics 31 (2005) 25–96. PDF
• Tautly foliated manifolds without R-covered foliations. Proceedings of the conference on Foliations and Dynamics, Warsaw 2000. This paper uses the constructions and techniques of "Graph manifolds and taut foliations" to show that there are tautly foliated graph manifolds which do not admit R-covered foliation. The manifolds all do, however, have finite covers which admit R-covered foliations. PDF
• Free Seifert surfaces and disk decompositions. Math. Zeit, 240 (2002) 197–210. This paper uses the construction of free genus one knots given in another paper, and work of Goda, to construct families of knots with genus one free Seifert surfaces which are not disk decomposable. PDF (color) PDF (black and white)
• Free genus one knots with large volume. Pacific J. Math. 201 (2001) 61–82. In this paper we construct a family of hyperbolic knots with free genus one (i.e, they each have a Seifert surface whose complement is a handlebody) whose complements have arbitrarily large volume. Together with the previous paper, these give examples of hyperbolic knots with free genus one and arbitrarily large canonical genus. These also provide examples of knots with an incompressible free Seifert surface which cannot be obtained from Seifert's algorithm applied to a projection of the knot. PDF
• Bounding canonical genus bounds volume. The canonical genus of a knot K is the minimum of the genera of Seifert surfaces built by Seifert's algorithm, taken over all projections of the knot K. In this paper we show for any g there is a constant C(g) so that any hyperbolic knot with canonical genus g has volume less than C(g). The bound on volume can in fact be chosen to be linear in g; in this paper we give a bound of 122g. PDF
• When incompressible tori meet essential laminations. (with R. Roberts) Pacific J. Math. 190 (1999) 21–40. In this paper we extend to essential laminations results on isotoping taut foliations. An essential lamination can always be isotoped so that it meets an incompressible torus tautly, i.e., the lamination remains essential after splitting the ambient manifold open along the torus, unless the lamination contains a cylindrical component; a pair of parallel torus leaves, with a collection of Reeb' annuli lying in between. This result plays a central role in the characterization of essential laminations in graph manifolds. PDF
• Persistent laminations from Seifert surfaces. J. Knot Thy. Ram 10 (2001) 1155–1168. In this paper we give a simple construction of persistent laminations in many knot complements, obtained by constructing a branched surface (and knot) from the Seifert surface of another knot. We show that the complement of the branched surface is essentially the same as the complement of the Seifert surface, so if one starts with an incompressible Seifert surface, one obtains an essential branched surface. Thus far the construction provides persistent laminations for 40 percent of the knots in the standard tables. PDF (with figures) You can find a list of the knots that have so far been built by this procedure, current as of July, 1998. You the reader are of course welcome to add to this list, by experimenting on your own; you can email me information about your discoveries. The SnapPea readable files for the knots so far constructed can be downloaded in Binhexed Stuffit, or just Stuffit form. There is also now a zipped archive of them (which is actually probably the most up-to-date).
• Persistently laminar tangles. J. Knot Theory and its Ramifications 8 (1999) 415–428. In this paper we show that an example of an essential lamination in the complement of the Stevedore's knot 6_1, due to Ulrich Oertel, can be associated to a certain tangle T_0, in a very strong way; the lamination remains essential in the complement of any knot K obtained by tangle sum with T_0. Even more, the lamination is persistent for K; it remains essential under every non-trivial Dehn filling along K. We also show how the construction generalizes to many more n-strand tangles. PDF
• Essential laminations, exceptional Seifert-fibered spaces, and Dehn filling. J. Knot Thy. Ram. 7 (1998) 425–432. In this paper we show how essential laminations can be used to provide an improvement on (some of) the results of the well-known 2pi-Theorem; we show that at most 20 Dehn fillings on a hyperbolic 3-manifold with boundary a torus T can yield a (reducible or finite pi_1 manifold or) small Seifert fibered space. The 2pi-Theorem gave a bound of 24.
• The classification of Dehn surgery on 2-bridge knots. (with Y.-Q. Wu) Comm. Anal. Geom. 9 (2001) 97–113. In this paper we complete the work of a previous paper, by showing that among 2-bridge knots, only torus knots and twist knots can admit a Dehn surgery which is a small Seifert-fibered space. This leads to a complete classification of surgeries on 2-bridge knots, according to whether the resulting manifold is finite pi_1, reducible, toroidal, seifert-fibered, or hyperbolic. PDF (with figures)
• Graph manifolds and taut foliations (with R. Naimi and R. Roberts). J. Diff. Geom. 45 (1997) 446–470. In this paper we examine the existence of foliations without Reeb components, taut foliations, foliations with no torus leaves, and Anosov flows, among graph manifolds. We show that each condition is strictly stronger than its predecessor(s), in the strongest possible sense; there are manifolds admitting foliations of each type which do not admit foliations of the succeeding type(s). PDF
• Essential laminations in Seifert-fibered spaces: Boundary behavior. Topology Appl. 95 (1999) 47–62. We show that, except for three specific manifolds M, an essential lamination in a Seifert-fibered space M with non-empty boundary cannot meet the boundary in a lamination with non-vertical Reeb annuli. As a corollary, any essential lamination in a torus knot exterior is (with a single exception) isotopic to one which is everywhere transverse to the foliation of the exterior by circles. This paper (together with other papers described here) finishes the topological characterization of essential laminations in Seifert-fibered spaces. PDF
• Small Seifert-fibered spaces and Dehn surgery on 2-bridge knots. Topology 37 (1998) 665–672. By combining the topological characterization of essential laminations in Seifert-fibered spaces, and constructions of Delman, we show that non-integer Dehn surgery on a (non-torus) 2-bridge knot never yields a small Seifert-fibered space. In most cases, no non-trivial surgery can yield one. PDF
• Essential laminations in I-bundles. Trans. AMS 349(1997) 1463-1485. In this paper we show that an essential lamination in an I-bundle over a closed surface can, with some well-known exceptions, be isotoped to lie everywhere transverse to the I-fibers. This result, which parallels some of the results of the Seifert-fibered space paper, is proved using the standard cell decomposition of an I-bundle and uses the Haken normal form techniques of the Haken normal form paper. PDF
• pi_1-injective, proper maps of open surfaces. The main result of this paper is an analogue of Nielsen's theorem for compact surfaces : a pi_1-injective, proper map of open, orientable surfaces either has degree zero and can be properly homotoped off of any compact subset of the range, or has non-zero degree, and can be properly homotoped to a finite-sheeted covering map.
• Essential laminations and deformations of homotopy equivalences: The structure of pullbacks. In this paper we study the structure of the inverse image of an essential lamination L under a homotopy equivalence of non-Haken 3-manifolds. We show that the tightly-wrapped' property of the essential lamination L (described in "Essential laminations in non-Haken manifolds") is in large part inherited by its pullback. PDF
• Essential laminations and deformations of homotopy equivalences: From essential pullback to homeomorphism. Topology and its Applications 60 (1994) 249–265. The main result of this paper is that if we have a homotopy equivalence f from M to N, where M is irreducible and N contains an essential lamination L such that f is transverse to L and the inverse image homotopic to a homeomorphism. This constitutes half of a program to show that, in the presence of an essential lamination, homotopy equivalent 3-manifolds are homeomorphic. PDF
• Essential laminations in non-Haken 3-manifolds. Topology and its Applications 53 (1993) 317-324. In this paper we show that an essential lamination in a non-Haken 3-manifold M is tightly-wrapped' - any two leaves have intersecting closures. We also show that this phenomenon holds for any lift of the essential lamination to a finite covering, thereby showing that tightly-wrappedness' cannot be used to detect a finite covering of M which is Haken. PDF
• Essential laminations and Haken normal form: Laminations with no holonomy. Communications in Analysis and Geometry 3 (1995) 465–477. The main result of this paper is that, given an essential lamination which has no holonomy, the infinite sequence of isotopies of the previous two papers are really finite sequences of isotopies. Consequently, the original lamination can be put into normal form. In the process a better understanding of how the infinite sequences of isotopies generally fail to terminate in finite time is also achieved. PDF
• Essential laminations and Haken normal form: Regular cell decompositions. This paper extends the result of the previous one to regular cell decompositions. The technique involves proving a similar convergence result, using an infinite sequence of infinite sequences of isotopies. PDF
• Essential laminations and Haken normal form. Pac. J. Math. 168 no. 2 (1995) 217–234. In this paper we show that, given an essential lamination in a 3-manifold M and a triangulation \tau of M, we can find a (possibly different) essential lamination which is in Haken normal form with respect to \tau. The technique is to build an infinite sequence of isotopies of the essential lamination, and to show that these isotopies converge' to a new lamination, which is in normal form. Some sublamination of the new lamination will be essential. PDF
• Essential laminations in Seifert-fibered spaces. Topology 32 no. 1 (1993) 61–85. We show that an essential lamination in a Seifert-fibered space M always contains a sublamination which can be made either horizontal' or `vertical' with respect to the foliation of M by circles. As a consequence, we find the first (and, to date, only) examples of 3-manifolds with universal cover R^3 which do not contain any essential laminations. PDF