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We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants.
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C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Gillers invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. After modifying a 2-knot into a Montesinos twin in a natural way, we show that Gillers invariant is related to the Seiberg-Witten invariant of the exterior of the twin, glued to the complement of a fiber in E(2).
We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.
Crowley and Nordstr{o}m introduced an invariant of $G_2$-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this invariant due to Crowley, Goette and Nordstr{o}m, we compute it for many of the closed torsion-free $G_2$-manifolds defined by Joyces generalized Kummer construction.
The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations of surfaces. This paper proves that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus $geq1$, when the approximating triangle meshes are reasonably good. To the best of the authors knowledge, this is the first convergence result on computing uniformizations for surfaces of genus $>1$.
Froyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in three and four-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of minimal hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Froyshov invariants of minimal $L$-spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Froyshov invariants for all spin$^c$ structures on the Seifert-Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.
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