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Research Papers
Dave Futer
- Cusp areas of Farey manifolds and applications to knot theory. With Effie Kalfagianni and
Jessica Purcell.
Submitted (2008).
[PS],
[PDF],
[ArXiv].
Abstract: We find explicit, combinatorial estimates for the cusp areas of
once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements.
Applications include volume estimates for the hyperbolic 3-manifolds obtained by Dehn
filling these bundles, for example estimates on the volume of closed 3-braid complements
in terms of the complexity of the braid word. We also relate the volume of a closed 3-braid
to certain coefficients of its Jones polynomial.
- Symmetric links and Conway sums: volume and Jones polynomial. With Effie Kalfagianni and
Jessica Purcell.
Submitted (2008).
[PS],
[PDF],
[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,
Xiao-Song Lin, and
Neal Stoltzfus. Submitted (2007).
[PS],
[PDF],
[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
quasi-trees of genus j of the dessin of a non-alternating 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,
Xiao-Song Lin, and
Neal Stoltzfus.
Journal of Combinatorial Theory, Series B 98 (2008), Issue 2, 384-399.
[PDF],
[Web],
[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
Bollobas-Riordan-Tutte 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 Bollobas-Riordan-Tutte
polynomial of a certain dessin associated to a link projection. We give some applications of this approach.
- Dehn filling, volume, and the Jones polynomial. With
Effie Kalfagianni and
Jessica Purcell. Journal of
Differential Geometry 78 (2008) 429-464.
[PS],
[PDF],
[ArXiv].
Abstract: Given a hyperbolic 3-manifold 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.
- Angled decompositions of arborescent link complements. With
François Guéritaud.
To appear in Proceedings of the London Mathematical Society (2008).
[PDF],
[Web],
[ArXiv].
Abstract: This paper describes a way to subdivide a 3-manifold 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.
- Geometric triangulations of two-bridge link complements.
Appendix to a paper by François Guéritaud.
Geometry & Topology 10 (2006), 1267-1282.
[PDF],
[Web],
[ArXiv].
Abstract: The complements of two-bridge links in S3 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), No. 3, 629-664.
[PDF],
[Web],
[ArXiv].
Abstract: If Thurston's Geometrization Conjecture is
true, then a closed 3-manifold is hyperbolic whenever it satisfies a
topological condition, called "hyperbolike". This paper proves a
mild diagrammatic condition on a knot or link in S3
under which any
non-trivial Dehn filling gives a hyperbolike closed manifold. For a
knot K, a non-trivial Dehn filling of K will be hyperbolike
whenever a prime, twist-reduced 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), No. 6, 741-748.
[PDF],
[Web],
[ArXiv].
Abstract: Let K be a knot that has an unknotting tunnel
tau. This paper proves that K admits a strong involution that
fixes tau pointwise if and only if K is a two-bridge knot and
tau its upper or lower tunnel. One result obtained along the way is
a version of the Smith conjecture for handlebodies: the fixed-point
set of an orientation-preserving, periodic diffeomorphism of a
handlebody is either empty or boundary-parallel.
- Cost-minimizing networks among immiscible fluids in R2. With
Andrei Gnepp, David McMath,
Brian Munson,
Ting Ng,
Sang-Hyoun Pahk, and Cara Yoder.
Pacific Journal of Mathematics 196 (2000), no. 2, 395-414.
[PS],
[PDF],
[Web].
Abstract: We model interfaces between immiscible fluids
as cost-minimizing 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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dfuter at math msu edu
Last modified: Mon Jul 9 11:15:42 PDT 2007
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