(10) Geometry & Topology 17 (2013) 925-974 On the equivalence of Legendrian and transverse invariants in knot Floer homology by John A Baldwin, David Shea Vela-Vick and Vera Vertesi URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p022.xhtml DOI: 10.2140/gt.2013.17.925
(1) Context-free manifold calculus and the Fulton-MacPherson operad by Victor Turchin
The paper gives an explicit description of the Weiss embedding tower in terms of spaces of maps of truncated modules over the framed Fulton-MacPherson operad.
(2) Restricting the topology of 1-cusped arithmetic 3-manifolds by Mark D Baker and Alan W Reid
This paper makes progress on classifying those closed orientable 3-manifolds M that contain knots K so that M \ K is arithmetic.
(3) The simplicial boundary of a CAT(0) cube complex by Mark F Hagen
For a CAT(0) cube complex X, we define a simplicial flag complex d_\triangle X, called the simplicial boundary, which is a natural setting for studying nonhyperbolic behavior of X. We compare d_\triangle X to the Roller, visual and Tits boundaries of X, give conditions under which the natural CAT(1) metric on d_\triangle X makes it isometric to the Tits boundary, and prove a more general statement relating the simplicial and Tits boundaries. The simplicial flag complex d_\triangle X allows us to interpolate between studying geodesic rays in X and the geometry of its contact graph \Gamma X, which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in X using d_\triangle X. Finally, we rephrase the rank-rigidity theorem of Caprace and Sageev in terms of group actions on \Gamma X and d_\triangle X and state characterizations of cubulated groups with linear divergence in terms of \Gamma X and d_\triangle X.
(4) Lipschitz minimality of Hopf fibrations and Hopf vector fields by Dennis DeTurck, Herman Gluck and Peter Storm
Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibers as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success.
(5) The classification of rational subtangle replacements between rational tangles by Kenneth L Baker and Dorothy Buck
A natural generalization of a crossing change is a rational subtangle replacement (RSR). We characterize the fundamental situation of the rational tangles obtained from a given rational tangle via RSR, building on work of Berge and Gabai, and determine the sites where these RSR may occur. In addition we also determine the sites for RSR distance at least two between 2-bridge links. These proofs depend on the geometry of the branched double cover. Furthermore, we classify all knots in lens spaces whose exteriors are generalized Seifert fibered spaces and their lens space surgeries, extending work of Darcy-Sumners. This work is in part motivated by the common biological situation of proteins cutting, rearranging and resealing DNA segments, effectively performing RSR on DNA `tangles'.
(6) Odd Khovanov homology by Peter S Ozsvath, Jacob Rasmussen and Zoltan Szabo
We describe an invariant of links in S^3 which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over Q. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.
(7) Faithful simple objects, orders and gradings of fusion categories by Sonia Natale
We establish some relations between the orders of simple objects in a fusion category and the structure of its universal grading group. We consider fusion categories that have a faithful simple object and show that their universal grading groups must be cyclic. As for the converse, we prove that a braided nilpotent fusion category with cyclic universal grading group always has a faithful simple object. We study the universal grading of fusion categories with generalized Tambara-Yamagami fusion rules. As an application, we classify modular categories in this class and describe the modularizations of braided Tambara-Yamagami fusion categories.
(8) Saturated fusion systems as idempotents in the double Burnside ring by Kari Ragnarsson and Radu Stancu
We give a new characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation. Interpreting our results in stable homotopy, we characterize the stable summands of the classifying space of a finite p-group that have the homotopy type of the classifying spectrum of a saturated fusion system, and prove an invariant theorem for double Burnside modules analogous to the Adams-Wilkerson criterion for rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Haynes Miller that proposes p-tract groups as a purely homotopy-theoretical model for p-local finite groups. We show that a p-tract group gives rise to a p-local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.
(9) On the number of ends of rank one locally symmetric spaces by Matthew Stover
Let Y be a noncompact rank one locally symmetric space of finite volume. Then Y has a finite number e(Y) > 0 of topological ends. In this paper, we show that for any natural number n, the Y with e(Y) at most n that are arithmetic fall into finitely many commensurability classes. In particular, there is a constant c_n such that n--cusped arithmetic orbifolds do not exist in dimension greater than c_n. We make this explicit for one-cusped arithmetic hyperbolic n-orbifolds and prove that none exist for n at least 30.
(10) On the equivalence of Legendrian and transverse invariants in knot Floer homology by John A Baldwin, David Shea Vela-Vick and Vera Vertesi
Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.
(11) Knot contact homology by Tobias Ekholm, John B Etnyre, Lenhard Ng and Michael G Sullivan
The conormal lift of a link K in R^3 is a Legendrian submanifold Lambda_K in the unit cotangent bundle U^* R^3 of R^3 with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of K, is defined as the Legendrian homology of Lambda_K, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization R x U^*R^3 with Lagrangian boundary condition R x Lambda_K. We perform an explicit and complete computation of the Legendrian homology of Lambda_K for arbitrary links K in terms of a braid presentation of K, confirming a conjecture that this invariant agrees with a previously defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot, which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.
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