[13] The classification of rational subtangle replacements between rational tangles(Joint
with Dorothy Buck);
submitted
Furthering the previous work, we use
Berge's classification of surgeries on knots in solid tori producing
solid tori to describe which pairs rational tangles are related by an
integral rational subtangle replacement (RSR). Also included is a
discussion of the "Berge tangle", a comparison with RSR between two
bridge links, and a classification of generalized Seifert fibered knots
in lens spaces and their lens space Dehn surgeries.
[12] Taxonomy of DNA Conformations within Complex Nucleoprotein Assemblies (Joint
with Dorothy Buck);
To appear in Progress of Theoretical Physics
Certain proteins chop-rearrange-glue
strands of DNA. Modeling this as rational subtangle replacements
between rational tangles we provide a classification of non-integral
replacements.
Here's my talk at the
Southeastern AMS meetings in Baton Rouge. It illustrates the
connections between grid diagrams for links in R3, S3, and lens spaces (other than S1x S2)
and Legendrian realizations of those links with respect to the
standard tight contact structures, i.e. the universally tight contact
structures. (By taking a standard contact 1-form on S3
that behaves well with the covering maps and then viewing lens spaces
up to contactomorphism, one may reckon with co-orientations.)
The powerpoint has a few animations (on the first few that talk about
legendrian curves and the very last slide) that should occur when you
click on the picutre. The pdf isn't animated.
These are the slides used for my talk at the national AMS meetings in San Diego. It gives a quick introduction
to small Seifert fibered spaces and the approach taken in a forthcoming
article to study the possibily of a large distance surgeries producing
them. They were drawn on paper with black pen and colored pencil,
then scanned. Shrinking the resolution to a smaller file size
for download somehow nuked the title page.
We show, in analogy with knots in S^3, that every knot in a lens space admits a grid diagram.
Consequentially the knot Floer homology of any knot in a lens space may be combinatorially computed.
These are the slides from
a talk given in Zacatecas, Mexico for the joint AMS-SMM meetings of May
2007. This is an overview of the works [5] and [7] below with a focus
on the resulting classification of once-punctured torus bundles that
admit lens space fillings. The figures in this talk were created with
XFig and Google
SketchUp. (It's free!)
Here
is the SketchUp file that illustrates the correspondence between the
Figure Eight knot and a certain closed three string braid presentation
of the unknot.
We determine the non-null
homologous knots in lens spaces whose exteriors contain properly
embedded once-punctured tori. All such knots arise as surgeries on the
Whitehead link and are grid number 1 in their lens spaces.
Communications in Analysis and
Geometry, 2009, vol 17, no 1
We determine the genus one
fibered knots in lens spaces that have tunnel number one. We also show
that every tunnel number one, once-punctured torus bundle is the result
of Dehn filling a component of the Whitehead link in the 3-sphere.
The braid axis of a closed
3-braid
lifts to a genus one fibered knot in the double cover of S^3 branched over
the closed braid. Every genus one fibered knot in a 3-manifold
may be obtained in this way. Using this perspective we answer
a
question of Morimoto about the number of genus one fibered knots in
lens spaces. We determine the number of genus one fibered
knots
up to homeomorphism in any given lens space. This number is 3 in the case of
the lens space L(4,1),
2 for the
lens spaces L(m,1)
with m>0,
and at most 1
otherwise.
We construct an algorithm
that lists all closed essential surfaces in the
complement of a knot that lies on the fiber of a trefoil or figure
eight knot.
Such knots are Berge knots and hence admit lens space surgeries.
Furthermore
they may have arbitrarily large hyperbolic volume. Using this algorithm
we
concoct large volume Berge knots of two flavors: those whose complement
contains arbitrarily many distinct closed essential surfaces, and those
whose
complement contains no closed essential surfaces.
Journal of Knot Theory and its Ramifications; 2008, September; pp 1099-1120
Using Kirby Calculus, we
explicitly pass from Berge's R-R descriptions of ten
families of knots with lens space surgeries to surgery descriptions on
the
minimally twisted five chain link (MT5C). Since the MT5C admits a
strong
involution, we also give the corresponding tangle descriptions.
Journal of Knot Theory and its Ramifications; 2008, September; pp 1077-1097
By obtaining surgery
descriptions of knots which lie on the genus one fiber
of the trefoil or figure eight knot, we show that these include
hyperbolic
knots with arbitrarily large volume. These knots admit lens space
surgeries and
form two families of Berge knots. By way of tangle descriptions we also
obtain
surgery descriptions for these knots on minimally twisted chain links.
In a lens space X of order r a knot K
representing an element of the fundamental group pi_1 X = Z/rZ of
order s \leq r
contains an orientable surface S properly embedded
in
its exterior X-N(K)
such that \bdry
S intersects the meridian of K
minimally s times.
Assume S
has just one boundary component. Let g be the minimal
genus of such surfaces
for K,
and assume s
\geq 4g-1. Then with respect to the genus
one Heegaard splitting of X, K has bridge number
at most 1.
We study knots that lie
as essential simple closed curves on the fiber of a genus one fibered
knot
in S^3.
We determine certain surgery descriptions of these knots
that enable estimates on volumes of these knots. We also
develop an
algorithm to list all closed essential surfaces in the complement of a
given knot in this family. Relationships between the volumes
of such
knots and the surfaces in their exteriors is then examined.
Preprints [2], [3], and [4] are derived from the dissertation and
include some expansions and corrections.
In preparation:
[#] Tunnel
number versus Heegaard genus - working title - (Joint
with Cameron Gordon and John Luecke)
Let K be a hyperbolic
knot in a hyperbolic manifold M
of Heegaard genus g
admitting a non-longitudinal S^3
surgery. Put K
into thin position with respect to a genus g splitting of M. There
exists a function w
such that K
intersects a thick level at most w(g)
times. As a consequence, if M is non-Haken, the
tunnel number of K
is at most w(g)/2+g-1.
[#] Small
Seifert fiber spaces and surgery - working title
- (Joint
with Cameron Gordon and John Luecke)
Let K be a hyperbolic
knot in S^3.
If non-integral, non-half-integral surgery on K produces a
non-Haken, Heegaard genus 2
manifold (in particular, a small Seifert fiber space) then K has tunnel number
at most 2.
Most of the figures in my
papers are created with XFig
(alt), an
open source vector
graphics editor. One nice thing about XFig is its ability to
export figures with text that can be rendered by LaTeX. This
pdf
gives a quick description of how to do this. The file XFigLaTeX.tar.gz contains
the LaTeX source, the original XFig file, and the two files exported
from XFig for LaTeX needed to produce the pdf.
(There are other ways of using XFig with LaTeX detailed here.
What I describe in the pdf is basically the approach "Type
C".)
