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Research Papers

Dave Futer

  1. Essential surfaces in highly twisted link complements.
    With Ryan Blair and Maggy Tomova.
    Algebraic & Geometric Topology, to appear. 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 n-punctured sphere visible in the diagram.

  2. Hyperbolic semi-adequate links.
    With Effie Kalfagianni and Jessica Purcell.
    Communications in Analysis & Geometry, to appear. ArXiv.

    Abstract: We provide a diagrammatic criterion for semi-adequate links to be hyperbolic. We also give a conjectural description of the satellite structures of semi-adequate links. One application of our result is that the closures of sufficiently complicated positive braids are hyperbolic links.

  3. Quasifuchsian state surfaces.
    With Effie Kalfagianni and Jessica Purcell.
    Transactions of the American Mathematical Society 366 (2014), Issue 8, 4323-4343. 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 graph--theoretic criterion in terms of a certain spine of the surfaces. For links with A- or B-adequate diagrams, the geometric type of the surface is also completely determined by a coefficient of the colored Jones polynomial of the link.

  4. Cusp geometry of fibered 3-manifolds.
    With Saul Schleimer.
    American Journal of Mathematics 136 (2014), Issue 2, 309-356. ArXiv.

    Abstract: Let F be a surface and suppose that φ : F → F is a pseudo-Anosov homeomorphism fixing a puncture p of F. The mapping torus M = Mφ 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 quasi-Fuchsian 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.

  5. Small volume link orbifolds.
    With Chris Atkinson.
    Mathematical Research Letters 20 (2013), Issue 6, 995-1016. ArXiv.

    Abstract: This paper proves lower bounds on the volume of a hyperbolic 3-orbifold whose singular locus is a link. We identify the unique smallest volume orbifold whose singular locus is a knot or link in the 3-sphere, or more generally in a Z6 homology sphere. We also prove more general lower bounds under mild homological hypotheses.

  6. 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, 51-77. 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.

  7. Fiber detection for state surfaces.
    Algebraic & Geometric Topology 13 (2013), Issue 5, 2799-2807. ArXiv.

    Abstract: Every Kauffman state σ of a link diagram D(K) naturally defines a state surface Sσ whose boundary is K. For a homogeneous state σ, we show that K is a fibered link with fiber surface Sσ 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 next-to-last 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].

  8. 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 (A-adequacy), 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 (A-adequacy), 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.

  9. Dehn filling and the geometry of unknotting tunnels.
    With Daryl Cooper and Jessica Purcell.
    Geometry & Topology 17 (2013), Issue 3, 1815-1876. ArXiv.

    Abstract: Any one-cusped hyperbolic manifold M with an unknotting tunnel τ is obtained by Dehn filling a cusp of a two-cusped 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 long-standing questions posed by Adams, Sakuma, and Weeks.

    We also construct an explicit sequence of one-tunnel knots in S3, all of whose unknotting tunnels have length approaching infinity.

  10. Explicit Dehn filling and Heegaard splittings.
    With Jessica Purcell.
    Communications in Analysis and Geometry 21 (2013), Issue 3, 625-650. 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(2g-1), 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 Moriah-Rubinstein and Rieck-Sedgwick.

  11. Explicit angle structures for veering triangulations.
    With François Guéritaud.
    Algebraic & Geometric Topology 13 (2013), Issue 1, 205-235. ArXiv.

    Abstract: Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3-manifolds. 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.

  12. Surface quotients of hyperbolic buildings.
    With Anne Thomas.
    International Mathematics Research Notices 2012, Issue 2, 437-477. ArXiv.

    Abstract: Let Ip,v be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph Kv,v. We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice Γ in Aut(Ip,v) such that the quotient of Ip,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(Ip,v) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.

  13. Volume bounds for generalized twisted torus links.
    With Abhijit Champanerkar, Ilya Kofman, Walter Neumann, and Jessica Purcell.
    Mathematical Research Letters 18 (2011), Issue 6, 1097-1120. 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.

  14. From angled triangulations to hyperbolic structures.
    With François Guéritaud.
    Contemporary Mathematics 541 (2011), 159-182. ArXiv.

    Abstract: This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold 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 non-linear part. The solutions to the linear equations form a convex polytope A. The solution to the non-linear 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.

  15. Slopes and colored Jones polynomials of adequate knots.
    With Effie Kalfagianni and Jessica Purcell.
    Proceedings of the American Mathematical Society 139 (2011), Issue 5, 1889-1896. 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.

  16. On diagrammatic bounds of knot volumes and spectral invariants.
    With Effie Kalfagianni and Jessica Purcell.
    Geometriae Dedicata 147 (2010), 115-130. ArXiv.

    Abstract: In recent years, several families of hyperbolic knots have been shown to have both volume and λ1 (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 two-sided bounds on either the volume or λ1. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on λ1. 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.

  17. Finite surgeries on three-tangle pretzel knots.
    With Masaharu Ishikawa, Yuichi Kabaya, Thomas Mattman, and Koya Shimokawa.
    Algebraic & Geometric Topology 9 (2009), Issue 2, 743-771. 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 non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Earlier work by Mattman, combined with Agol and Lackenby's 6-theorem, reduces the argument to knots with small indices p,q,r. We treat these using the Culler-Shalen 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.

  18. Cusp areas of Farey manifolds and applications to knot theory.
    With Effie Kalfagianni and Jessica Purcell.
    International Mathematics Research Notices 2010, Issue 23, 4434-4497. 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.

  19. Symmetric links and Conway sums: volume and Jones polynomial.
    With Effie Kalfagianni and Jessica Purcell.
    Mathematical Research Letters 16 (2009), Issue 2, 233-253. 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.

  20. Alternating sum formulae for the determinant and other link invariants.
    With Oliver Dasbach, Effie Kalfagianni, Xiao-Song Lin, and Neal Stoltzfus.
    Journal of Knot Theory and its Ramifications 19 (2010), Issue 6, 765-782. 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.

  21. 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. 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.

  22. Angled decompositions of arborescent link complements.
    With François Guéritaud.
    Proceedings of the London Mathematical Society 98 (2009), Issue 2, 325-364. 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.

  23. Dehn filling, volume, and the Jones polynomial.
    With Effie Kalfagianni and Jessica Purcell.
    Journal of Differential Geometry 78 (2008), Issue 3, 429-464. 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.

  24. Geometric triangulations of two-bridge link complements.
    Appendix to a paper by François Guéritaud.
    Geometry & Topology 10 (2006), 1267-1282. 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.

  25. Links with no exceptional surgeries.
    With Jessica Purcell.
    Commentarii Mathematici Helvetici 82 (2007), Issue 3, 629-664. ArXiv.

    Abstract: If Thurston's Geometrization Conjecture is true, then a closed 3-manifold is hyperbolic whenever it satisfies a topological criterion, 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.

  26. Involutions of knots that fix unknotting tunnels.
    Journal of Knot Theory and its Ramifications 16 (2007), Issue 6, 741-748. 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 two-bridge knot and τ 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.

  27. Cost-minimizing networks among immiscible fluids in R2.
    With Andrei Gnepp, David McMath, Brian Munson, Ting Fai Ng, Sang-Hyoun Pahk, and Cara Yoder.
    Pacific Journal of Mathematics 196 (2000), Issue 2, 395-414.

    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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Last modified: Mon Jul 9 11:15:42 PDT 2007