
Research Papers
Dave Futer
 The lowest
volume 3orbifolds with high torsion.
With Chris
Atkinson.
Submitted (2015).
ArXiv.
Abstract: For each natural number n ≥ 4, we determine the unique
lowest volume hyperbolic 3orbifold whose torsion orders are bounded below by
n. This lowest volume orbifold has base space the 3sphere and singular locus
the figure8 knot, marked n. We apply this result to give sharp lower bounds
on the volume of a hyperbolic manifold in terms of the order of elements in
its symmetry group.
 Essential surfaces
in highly twisted link complements.
With Ryan Blair
and Maggy
Tomova.
Algebraic & Geometric Topology 15 (2015) 15011523.
ArXiv.
Abstract: We prove that in the complement of a highly twisted link, all
closed, essential, meridionally incompressible surfaces must have high
genus. The genus bound is proportional to the number of crossings per twist
region. A similar result holds for surfaces with meridional boundary: such a
surface either has large negative Euler characteristic, or is an npunctured
sphere visible in the diagram.
 Hyperbolic
semiadequate links.
With
Effie Kalfagianni and
Jessica Purcell.
Communications in Analysis & Geometry, to appear.
ArXiv.
Abstract: We provide a diagrammatic criterion for semiadequate links to
be hyperbolic. We also give a conjectural description of the satellite
structures of semiadequate links. One application of our result is that the
closures of sufficiently complicated positive braids are hyperbolic links.
 Quasifuchsian state
surfaces.
With
Effie Kalfagianni and
Jessica Purcell.
Transactions
of the American Mathematical Society 366 (2014), Issue 8,
43234343.
ArXiv.
Abstract: This paper continues our study, initiated in
the monograph [19], of essential state surfaces in link complements that
satisfy a mild diagrammatic hypothesis (homogeneously adequate). For
hyperbolic links, we show that the geometric type of these surfaces in the
Thurston trichotomy is completely determined by a simple graphtheoretic
criterion in terms of a certain spine of the surfaces. For links with
A or Badequate diagrams, the geometric type of the surface is
also completely determined by a coefficient of the colored Jones
polynomial of the link.
 Cusp geometry of fibered
3manifolds.
With Saul
Schleimer.
American
Journal of Mathematics 136 (2014), Issue 2, 309356.
ArXiv.
Abstract:
Let F be a surface and suppose that φ : F → F is a
pseudoAnosov homeomorphism fixing a puncture p of F. The
mapping torus M = M_{φ} is hyperbolic and contains a maximal cusp
C about the puncture p.
We show that the area and height of the cusp torus ∂C are equal,
up to explicit multiplicative error, to the stable translation
distance of φ acting on the arc complex A(F,p). Our proofs
rely on elementary facts about the hyperbolic geometry of pleated
surfaces. In particular, we do not use any deep results in
Teichmüller theory, Kleinian group theory, or the coarse geometry of
A(F,p).
A similar result holds for quasiFuchsian manifolds N ≈ F
× R. In that setting, we prove a combinatorial estimate on the
area and height of the cusp annulus in the convex core of N and give
explicit multiplicative and additive errors.
 Small volume link
orbifolds.
With Chris
Atkinson.
Mathematical Research Letters 20 (2013), Issue 6, 9951016.
ArXiv.
Abstract: This paper proves lower bounds on the volume of a hyperbolic
3orbifold whose singular locus is a link. We identify the unique smallest
volume orbifold whose singular locus is a knot or link in the 3sphere, or
more generally in a Z_{6} homology sphere. We also prove more general lower
bounds under mild homological hypotheses.
 Jones polynomials, volume, and
essential knot surfaces: a survey.
With Effie Kalfagianni and
Jessica Purcell.
Proceedings of
Knots in Poland III, Banach Center Publications 100
(2014), Issue 1, 5177.
ArXiv.
Abstract:
This paper is a brief overview of recent results by the authors relating
colored Jones polynomials to geometric topology. The proofs of these results
appear in the papers [13] and [19], while this
survey focuses on the main ideas and examples.
 Fiber detection for state
surfaces.
Algebraic &
Geometric Topology 13 (2013), Issue 5,
27992807.
ArXiv.
Abstract: Every Kauffman state σ of a link
diagram D(K) naturally defines a state
surface S_{σ} whose boundary is K. For a
homogeneous state σ, we show that K is a fibered link
with fiber surface S_{σ} if and only if an associated
graph G'_{σ} is a tree.
As a corollary, it follows that for an adequate knot or link, the second and
nexttolast coefficients of the Jones polynomial are obstructions to certain
state surfaces being fibers for K.
This provides a dramatically simpler proof of a theorem from monograph [20].
 Guts of surfaces and the colored
Jones polynomial.
With Effie Kalfagianni and
Jessica Purcell.
Monograph published
in Lecture
Notes in Mathematics (Springer),
volume 2069 (2013).
ArXiv.
Abstract:
This monograph derives direct and concrete relations between colored Jones
polynomials and the topology of incompressible spanning surfaces in knot and
link complements. Under mild diagrammatic hypotheses that arise naturally in
the study of knot polynomial invariants (Aadequacy), we prove that the growth
of the degree of the colored Jones polynomials is a boundary slope of an
essential surface in the knot complement. We show that certain coefficients of
the polynomial measure how far this surface is from being a fiber in the knot
complement; in particular, the surface is a fiber if and only if a particular
coefficient vanishes. Our results also yield concrete relations between
hyperbolic geometry and colored Jones polynomials: for certain families of
links, coefficients of the polynomials determine the hyperbolic volume to
within a factor of 4.
Our approach is to generalize the checkerboard decompositions of alternating
knots. Under mild diagrammatic hypotheses (Aadequacy), we show that the
checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral
decomposition of their complement. We employ normal surface theory to
establish
a dictionary between the pieces of the JSJ decomposition of the surface
complement and the combinatorial structure of certain spines of the
checkerboard surface (state graphs). Since state graphs have previously
appeared in the study of Jones polynomials, our setting and methods create a
bridge between quantum and geometric knot invariants.
 Dehn filling and the
geometry of unknotting tunnels.
With
Daryl Cooper and
Jessica Purcell.
Geometry &
Topology 17 (2013), Issue 3, 18151876.
ArXiv.
Abstract:
Any onecusped hyperbolic manifold M with an unknotting tunnel τ is obtained
by Dehn filling a cusp of a twocusped hyperbolic manifold.
In the case where M is obtained by "generic" Dehn filling, we prove that τ is isotopic to a geodesic,
and characterize whether τ is isotopic to an edge in the canonical
decomposition of M. We also give explicit estimates (with additive error only)
on the length of τ relative to a maximal cusp. These results give generic
answers to three longstanding questions posed by Adams, Sakuma, and Weeks.
We also construct an explicit sequence of onetunnel knots
in S^{3},
all of whose unknotting tunnels have length approaching infinity.
 Explicit Dehn filling and
Heegaard splittings.
With Jessica Purcell.
Communications
in Analysis and Geometry 21 (2013), Issue 3, 625650.
ArXiv.
Abstract: We prove an explicit, quantitative criterion that ensures the
Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped
hyperbolic manifold X, and a Dehn filling whose meridian and longitude curves
are longer than 2π(2g1), we show that every genus g Heegaard splitting of
the filled manifold comes from a splitting of the original manifold X. The
analogous statement holds for fillings of multiple boundary tori. This gives
an effective version of a theorem of MoriahRubinstein and RieckSedgwick.
 Explicit angle structures for veering
triangulations.
With
François
Guéritaud.
Algebraic &
Geometric Topology 13 (2013), Issue 1, 205235.
ArXiv.
Abstract:
Agol recently introduced the notion of a veering triangulation, and showed
that such triangulations naturally arise as layered triangulations of fibered
hyperbolic 3manifolds. We prove, by a constructive argument, that every
veering triangulation admits positive angle structures, recovering a result of
Hodgson, Rubinstein, Segerman, and Tillmann. Our construction leads to
explicit lower bounds on the smallest angle in this positive angle structure,
and to information about angled holonomy of the boundary tori.
 Surface quotients of
hyperbolic buildings.
With Anne Thomas.
International
Mathematics Research Notices 2012, Issue 2, 437477.
ArXiv.
Abstract:
Let I_{p,v} be Bourdon's building, the unique simplyconnected 2complex such
that all 2cells are regular rightangled hyperbolic pgons and the link at
each vertex is the complete bipartite graph K_{v,v}. We investigate and mostly
determine the set of triples (p,v,g) for which there exists a uniform lattice
Γ in Aut(I_{p,v}) such that the quotient of I_{p,v} by
Γ is a compact orientable surface of genus
g. Surprisingly, the existence of Γ depends upon the value of v. The
remaining cases lead to open questions in tessellations of surfaces and in
number theory. Our construction of Γ, together with a theorem of Haglund,
implies that for p ≥ 6, every uniform lattice in Aut(I_{p,v}) contains a surface
subgroup. We use elementary group theory, combinatorics, algebraic topology,
and number theory.
 Volume bounds for generalized
twisted torus links.
With
Abhijit Champanerkar,
Ilya Kofman,
Walter Neumann, and
Jessica Purcell.
Mathematical Research
Letters 18 (2011), Issue 6, 10971120.
ArXiv.
Abstract:
Twisted torus knots and links are given by twisting adjacent strands of a
torus link. They are geometrically simple and contain many examples of the
smallest volume hyperbolic knots. Many are also Lorenz links.
We study the geometry of twisted torus links and related generalizations. We
determine upper bounds on their hyperbolic volumes that depend only on the
number of strands being twisted. We exhibit a family of twisted torus knots
for which this upper bound is sharp, and another family with volumes
approaching infinity. Consequently, we show there exist twisted torus knots
with arbitrarily large braid index and yet bounded volume.
 From angled triangulations to
hyperbolic structures.
With François
Guéritaud.
Contemporary
Mathematics 541 (2011), 159182.
ArXiv.
Abstract:
This survey paper contains an elementary exposition of Casson and Rivin's
technique for finding the hyperbolic metric on a 3manifold M with toroidal
boundary. We also survey a number of applications of this technique.
The method involves subdividing M into ideal tetrahedra and solving a system
of gluing equations to find hyperbolic shapes for the tetrahedra. The gluing
equations decompose into a linear and nonlinear part. The solutions to the
linear equations form a convex polytope A. The solution to the nonlinear part
(unique if it exists) is a critical point of a certain volume functional on
this polytope. The main contribution of this paper is an elementary proof of
Rivin's theorem that a critical point of the volume functional on A produces a
complete hyperbolic structure on M.
 Slopes and colored Jones
polynomials of adequate knots.
With Effie Kalfagianni and
Jessica Purcell.
Proceedings
of the American Mathematical Society 139 (2011), Issue 5,
18891896.
ArXiv.
Abstract:
Garoufalidis conjectured a relation between the boundary slopes of a
knot and its colored Jones polynomials. According to the conjecture,
certain boundary slopes
are detected by the sequence of degrees of the colored Jones polynomials.
We verify this conjecture for adequate knots, a class
that vastly generalizes that of alternating knots.
 On diagrammatic bounds of knot
volumes and spectral invariants.
With Effie Kalfagianni and
Jessica Purcell.
Geometriae
Dedicata 147 (2010), 115130.
ArXiv.
Abstract:
In recent years, several families of hyperbolic knots have been shown to have
both volume and λ_{1} (first eigenvalue of the Laplacian) bounded in
terms of the twist number of a diagram, while other families of knots have
volume bounded by a generalized twist number. We show that for general knots,
neither the twist number nor the generalized twist number of a diagram can
provide twosided bounds on either the volume or λ_{1}. We do so by
studying the geometry of a family of hyperbolic knots that we call double coil
knots, and finding twosided bounds in terms of the knot diagrams on both the
volume and on λ_{1}. We also extend a result of Lackenby to show that a
collection of double coil knot complements forms an expanding family iff their volume
is bounded.
 Finite surgeries on
threetangle pretzel knots.
With Masaharu
Ishikawa,
Yuichi Kabaya,
Thomas Mattman, and
Koya
Shimokawa.
Algebraic &
Geometric Topology 9 (2009), Issue 2, 743771.
ArXiv.
Abstract: We classify Dehn surgeries on (p,q,r) pretzel knots that
result in a manifold of finite fundamental group. The only hyperbolic pretzel
knots that admit nontrivial finite surgeries are (2,3,7) and
(2,3,9). Earlier work by Mattman, combined with Agol
and Lackenby's 6theorem, reduces the argument to knots with small indices
p,q,r. We treat these using the CullerShalen norm of the SL(2,C)character
variety. In particular, we introduce new techniques for demonstrating that
boundary slopes are detected by the character variety.
 Cusp areas of Farey
manifolds and applications to knot theory.
With Effie Kalfagianni and
Jessica Purcell.
International
Mathematics Research Notices
2010, Issue 23, 44344497.
ArXiv.
Abstract: We find explicit, combinatorial estimates for the cusp areas of
oncepunctured torus bundles, 4punctured sphere bundles, and 2bridge link complements.
Applications include volume estimates for the hyperbolic 3manifolds obtained by Dehn
filling these bundles, for example estimates on the volume of closed 3braid complements
in terms of the complexity of the braid word. We also relate the volume of a closed 3braid
to certain coefficients of its Jones polynomial.
 Symmetric links and Conway sums:
volume and Jones polynomial.
With Effie Kalfagianni and
Jessica Purcell.
Mathematical Research
Letters 16 (2009), Issue 2, 233253.
ArXiv.
Abstract: We obtain bounds on hyperbolic volume for periodic links and Conway sums of
alternating tangles. For links that are Conway sums we also bound the hyperbolic volume
in terms of the coefficients of the Jones polynomial.
 Alternating sum formulae for the
determinant and other link invariants.
With
Oliver Dasbach,
Effie Kalfagianni,
XiaoSong Lin, and
Neal Stoltzfus.
Journal
of Knot Theory and its Ramifications
19 (2010), Issue 6, 765782.
ArXiv.
Abstract: A classical result states that the determinant of an alternating link is equal
to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link.
We generalize this result to show that the determinant is the alternating sum of the number of
quasitrees of genus j of the dessin of a nonalternating link.
Furthermore, we obtain formulas for other link invariants by counting quantities on dessins.
 The Jones polynomial and
graphs on surfaces.
With
Oliver Dasbach,
Effie Kalfagianni,
XiaoSong Lin, and
Neal Stoltzfus.
Journal of
Combinatorial Theory, Series B 98 (2008), Issue 2, 384399.
ArXiv.
Abstract: The Jones polynomial of an alternating link is a certain specialization of the Tutte
polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The
BollobasRiordanTutte polynomial generalizes the Tutte plolynomial of planar graphs to graphs that are
embedded in closed surfaces of higher genus (i.e. dessins d'enfant).
In this paper we show that the Jones polynomial of any link can be obtained from the BollobasRiordanTutte
polynomial of a certain dessin associated to a link projection. We give some applications of this approach.
 Angled decompositions of arborescent
link complements.
With François
Guéritaud.
Proceedings
of the London Mathematical Society 98 (2009), Issue 2, 325364.
ArXiv.
Abstract: This paper describes a way to subdivide a 3manifold into angled blocks, namely
polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral
angles that fit together in a consistent fashion, we prove that a manifold constructed from
these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished
theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families
of exceptions, have hyperbolic complements.
 Dehn filling, volume, and the
Jones polynomial.
With
Effie Kalfagianni and
Jessica Purcell.
Journal of
Differential Geometry 78 (2008), Issue 3, 429464.
ArXiv.
Abstract: Given a hyperbolic 3manifold with torus boundary, we bound the
change in volume under a Dehn filling where all slopes have length
at least 2π. This result is applied to give explicit diagrammatic
bounds on the volumes of many knots and links, as well as
their Dehn fillings and branched covers. Finally, we use this
result to bound the volumes
of knots in terms of the coefficients of their Jones polynomials.
 Geometric
triangulations of twobridge link complements.
Appendix to a paper by
François
Guéritaud.
Geometry &
Topology 10 (2006), 12671282.
ArXiv.
Abstract: The complements of twobridge links in S^{3} have a natural decomposition into
topological ideal tetrahedra, described by Sakuma and Weeks. Following the lead of Guéritaud, we use
volume maximization techniques to give this ideal triangulation a complete hyperbolic structure. Applications
of this method include sharp volume estimates and a result (conjectured by Thistlethwaite) about arcs in the
projection plane being hyperbolic geodesics.
 Links with no
exceptional surgeries.
With
Jessica Purcell.
Commentarii
Mathematici Helvetici 82 (2007), Issue 3, 629664.
ArXiv.
Abstract: If Thurston's Geometrization Conjecture is
true, then a closed 3manifold is hyperbolic whenever it satisfies a
topological criterion, called "hyperbolike". This paper proves a
mild diagrammatic condition on a knot or link in S^{3}
under which any
nontrivial Dehn filling gives a hyperbolike closed manifold. For a
knot K, a nontrivial Dehn filling of K will be hyperbolike
whenever a prime, twistreduced diagram of K has at least 4 twist
regions and at least 6 crossings per twist region; the statement for
links is similar.
We prove this result using two arguments, one geometric and one
combinatorial. The combinatorial argument also proves that every
link with at least 2 twist regions and at least 6 crossings per
twist region is hyperbolic and gives a lower bound for the genus of a link.
 Involutions of knots that fix
unknotting tunnels.
Journal
of Knot Theory and its Ramifications 16 (2007), Issue 6, 741748.
ArXiv.
Abstract: Let K be a knot that has an unknotting tunnel
τ. This paper proves that K admits a strong involution that
fixes tau pointwise if and only if K is a twobridge knot and
τ its upper or lower tunnel. One result obtained along the way is
a version of the Smith conjecture for handlebodies: the fixedpoint
set of an orientationpreserving, periodic diffeomorphism of a
handlebody is either empty or boundaryparallel.
 Costminimizing networks among
immiscible fluids in R^{2}.
With
Andrei Gnepp, David McMath,
Brian
Munson,
Ting Fai Ng, SangHyoun Pahk, and Cara Yoder.
Pacific Journal of
Mathematics 196 (2000), Issue 2, 395414.
Abstract: We model interfaces between immiscible fluids
as costminimizing networks, where "cost" is a weighted length. We
consider conjectured necessary and sufficient conditions for when a planar
cone is minimizing. In some cases we give a proof; in other cases we
provide a counterexample.
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Last modified: Mon Jul 9 11:15:42 PDT 2007
