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.
New method uses mosaics to study wild knots.
Proves minimal crossing diagrams for specific spatial graphs.
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial graphs and singular knots, that is obtained by replacing unknown crossings or ve…
New invariants distinguish spatial graphs not previously possible.
Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is simil…
The paper classifies palettes of Dehn colorings for spatial graphs.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split li…
New formulas for spatial 2-bouquet graphs discovered.
Algorithm determines spatial graph isomorphism with vertex, edge colorings and orientations.
The Kauffman-Vogel polynomials are three variable polynomial invariants of -valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented -valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with . Bataineh, Elha…
Enhances knot counting using mosaic diagrams.
A new mosaic system for immersed surface-links is introduced.
Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors deve…
This paper proposes a new Quantum Spatial Graph Convolutional Neural Network (QSGCNN) model that can directly learn a classification function for graphs of arbitrary sizes. Unlike state-of-the-art Graph Convolutional Neural Network (GCNN) models, the proposed QSGCNN model incorporates the process of identifying transit…
Graph Lie algebras have infinite prolongation if they have a vertex of degree one.
The paper studies graph products of groups and recovers graph and vertex groups under certain conditions.
Graphically discrete groups have strong rigidity properties.
Gaussian processes classify graphs using vertex and edge features.
A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We…
Explicit presentations found for asymptotically rigid mapping class groups.
Study graph products of groups, classifying them up to measure equivalence and rigidity.
Let be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group quasi-isometric to is abstractly commensurable with .
A natural approach to analyze interaction data of form "what-connects-to-what-when" is to create a time-series (or rather a sequence) of graphs through temporal discretization (bandwidth selection) and spatial discretization (vertex contraction). Such discretization together with non-negative factorization techniques c…
The paper explores new quandle systems for handlebody-links and spatial graphs.
The paper studies tunnel and bridge numbers of composite genus 2 spatial graphs.
We developed a convolution neural network (CNN) on semi-regular triangulated meshes whose vertices have 6 neighbours. The key blocks of the proposed CNN, including convolution and down-sampling, are directly defined in a vertex domain. By exploiting the ordering property of semi-regular meshes, the convolution is defin…
New proof for global rigidity of vertex scaling on polyhedral surfaces.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
We construct a series of finitely presented semigroups. The centers of these semigroups encode uniquely up to rigid ambient isotopy in 3-space all non-oriented spatial graphs. This encoding is obtained by using three-page embeddings of graphs into the product of the line with the cone on three points. By exploiting thr…
The paper estimates Betti numbers for graphs with specific curvatures, proving bounds and characterizing rigidity.
A ravel is a spatial graph which is non-planar but contains no non-trivial knots or links. We characterize when a Montesinos tangle can become a ravel as the result of vertex closure with and without replacing some number of crossings by vertices.
Paper introduces spherical knot mosaics for knot and link invariants.
Rectangular mosaics extend virtual knot studies to larger polygons.
In 2008, Kauffman and Lomonaco introduce the concepts of a knot mosaic and the mosaic number of a knot or link, the smallest integer such that a knot or link can be represented on an -mosaic. In arXiv:1702.06462, the authors explore space-efficient knot mosaics and the tile number of a knot or link, the smallest…
Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on `Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an matrix whose entries are eleven mosaic tiles, represent…
Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weig…
Researchers found algorithms to construct toric mosaics and set upper bounds for their numbers.
Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S^3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes wit…
Algorithm finds mosaic numbers for knots with 10 or fewer crossings.
Study uses knot theory to model RNA foldings, emphasizing both entanglement and intrachain interactions.
New rigidity result for hyperbolic surfaces based on curve lengths.
Expanding on prime knots with 6 or less mosaic tiles, this paper analyzes those with 7 tiles.
Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer such that the knot can be represented as a knot -mosaic. In this paper we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an -mosaic and any knot that…
This paper studies virtual knots using mosaic diagrams.
This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poinca…
Computes bounds on mosaic number of Legendrian knots.
Improved bounds for knot crossings in different mosaic patterns.