** Research papers and preprints**

For the most part, the papers archived here, where published, represent the last preprint version of the paper.

** Tame filling invariants for groups (with S. Hermiller)**, submitted.

Download as a PDF file.

** A uniform model for almost convexity and rewriting systems (with S. Hermiller)**, submitted.

Download as a PDF file.

*Algorithms and topology for Cayley graphs of groups* (with S. Hermiller and D. Holt)

to appear in J. Algebra.

Download as a PDF file.

*4-moves and the Dabkowski-Sahi invariant for knots* (with S. Hermiller and R. Todd)

J. Knot Theory Ramifications **22** (2013) 1350069.1-20.

Download as a PDF file.

*Subgroups of free idempotent generated semigroups: full linear monoid* (with Stuart Margolis and John Meakin)

arXiv:1009.5683

Download as a PDF file.

*Subgroups of free idempotent generated semigroups need not be free* (with S. Margolis and J. Meakin)

J. Algebra **321** (2009) 3026--3042.

Download as a PDF file.

*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.

Download the Postscript file (1150K)
or PDF file (285K) (with black and white figures - more printable),

or the Postscript file (2750K)
or PDF file (285K) (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.

Download the Postscript file (1150K)
or PDF file (285K).

*The Heegaard genus of bundles over S^1* (with Yo'av Rieck)

Geometry and Topology Monographs **12** (2007) 17--35.

Download as a PDF file.

*Canonical genus and the Whitehead doubles of families of alternating knots* (with Jacqueline Jensen)

math.GT/0608765

Download as a PDF file.

*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.

Download as a PDF file.

* 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.

Download Postscript file
or PDF file.

* 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.

The paper comes in two flavors: with figures in color (for viewing) and figures in
black and white (for printing). Download color version as a
Postscript file or
PDF file; download black and white version
as a Postscript file or
PDF file.

* 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.

Download as a Postscript file or
PDF file, both containing figures.

*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 .

Download as a
Postscript file or
PDF file, both containing figures.

(with R. Roberts) *When incompressible tori meet essential
laminations*

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, above.

Download Dvi file or
Postscript file or
PDF file.

*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.

Download as a
Dvi file (without figures), or as a
Postscript
or PDF file (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.

Download as a Dvi file (without figures), or as a
Postscript or
PDF file (with figures).

*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.

Download Dvi file or
Postscript file or
PDF file.

(with Y.-Q. Wu) *The classification of Dehn surgery on
2-bridge knots*

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.

Download Dvi file (without figures) or
Postscript file
or
PDF file (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).

Download Dvi file or
Postscript file
or
PDF file.

*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.

Download Dvi file or
Postscript file
or PDF file.

(An interesting story about the manuscript...)

*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.

Download Dvi file or
Postscript file
or
PDF file.

*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.

Download Dvi file or
Postscript file
or
PDF file.

*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.

Download Dvi file or
Postscript file
or
PDF file.

*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.

Download Dvi file or
Postscript file
or
PDF file.

*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.

Download Dvi file or
Postscript file
or
PDF file.

*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.

Download
Dvi file or
Postscript file
or
PDF file.

*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.

Download Dvi file or
Postscript file
or
PDF file.

*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.

Download Dvi file or
Postscript file
or
PDF file.

* 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.

Download Dvi file or
Postscript file
or
PDF file.

.