A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study counts geodesic surfaces in knot complements, finding unique ones for small knots.
The study identifies flat manifolds with unique cusp cross-sections in arithmetic hyperbolic manifolds.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing d…
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
Minimal crossing number found in arithmetic curve systems.
Paper proposes new density estimators for high-dimensional data.
Uniform proof reconstructs spaces using cross ratio on boundary.
A novel OT-based method for aligning hyperbolic representations.
Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.
We show that there exists a suitable neighborhood of a constant curvature hyperbolic metric such that, for all initial data in this neighborhood, the corresponding solution to a normalized cross curvature flow exists for all time and converges to a hyperbolic metric. We show that the same technique proves an analogous …
The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.
Improved bounds on volume-det inequality for links with many twists.
A well-known conjecture in knot theory says that the percentage of hyperbolic knots amongst all of the prime knots of or fewer crossings approaches as approaches infinity. In this paper, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjectur…
Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten mod…
Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cubulations of arbitrary Gromov hyperbolic groups . Und…
The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-me…
We show that the proportion of hyperbolic knots among all of the prime knots of or fewer crossings does not converge to as approaches infinity. Moreover, we show that if is a nontrivial knot then the proportion of satellites of among all of the prime knots of or fewer crossings does not converge…
We show that all closed flat n-manifolds are diffeomorphic to a cusp cross-section in a finite volume hyperbolic (n+1)-orbifold.
We introduce curvature-adapted foliations of complex hyperbolic space and study some of their properties. Generalized pseudo-Einstein hypersurfaces of complex hyperbolic space are classified. Analogous results for curvature-adapted hypersurfaces of quaternionic hyperbolic space are also obtained.
Study new conjectures linking knot volume and knot cohomology.
Various factorization-based methods have been proposed to leverage second-order, or higher-order cross features for boosting the performance of predictive models. They generally enumerate all the cross features under a predefined maximum order, and then identify useful feature interactions through model training, which…
We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing number. We also compute their Jones polynomials.
We use Korányi--Reimann complex cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of complex hyperbolic space and the first Heisenberg group.
We show that if is a nontrivial knot then the proportion of satellites of among all of the prime non-split links of or fewer crossings does not converge to as approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of or fewe…
For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient co…
Positive braids minimize knot untangling steps.
Cross-regularization adapts model complexity during training.
We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.
Study laws of cosines and sines for hyperbolic shapes with ideal vertices.
The paper bounds crossing numbers of dense graphs on surfaces.
Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link L into bipyramids, given any multi-crossing projection of L. When L is hyperbolic, this gives new upper bounds on the volume of L given its multi-crossing projection. These bounds are realized by three closely rela…
Study quantizes ropelength and writhe of 12-crossing knots.
Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.
The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action on R^3 by polyno…
Agent finds unknotting sequences for complex knots.
Nowadays, users open multiple accounts on social media platforms and e-commerce sites, expressing their personal preferences on different domains. However, users' behaviors change across domains, depending on the content that users interact with, such as movies, music, clothing and retail products. In this paper, we pr…
We tackle unsupervised anomaly detection (UAD), a problem of detecting data that significantly differ from normal data. UAD is typically solved by using density estimation. Recently, deep neural network (DNN)-based density estimators, such as Normalizing Flows, have been attracting attention. However, one of their draw…
We generalise in this article the Mc Shane-Mirzakhani identities in hyperbolic geometry to arbitrary cross ratios. We give an expression of them in the case of Hitchin representations of surface groups in PSL(n, R) in a suitable choice of Fock-Goncharov coordinates.
Given a reduced alternating diagram for a link, we obtain conditions that guarantee that the link complement has a complete hyperbolic structure, crossing arcs are the edges of an ideal geodesic triangulation, and every crossing arc is isotopic to a simple geodesic. The latter was conjectured by Sakuma and Weeks in 199…
Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists…
We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties …
A new model for imputing missing values in time series data across domains.
A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the we…
Cross-domain recommendation has long been one of the major topics in recommender systems. Recently, various deep models have been proposed to transfer the learned knowledge across domains, but most of them focus on extracting abstract transferable features from auxilliary contents, e.g., images and review texts, and th…
We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered ca…
Formula for Casson-Walker invariant; applies to knot complements and cosmetic crossing conjecture.
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(…