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.
Khovanov homology detects essential surfaces in knot complements.
Paper extends quantum invariant to colored ideal triangulations.
We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a,q) super-polynomial of all knots in 3-space. Our result has implications on the quantizatio…
The paper finds 3-colorings of 2-sphere triangulations.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
Combinatorial description of 3-manifolds using ordered triangulations.
Minimal ideal triangulations studied for hyperbolic 3-manifolds.
The paper extends trisection theory to non-orientable 4-manifolds using colored triangulations.
Study of colored triangulations linked to symmetric groups.
Essential loops found in taut ideal triangulations.
A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behavio…
In this paper we prove that any triangulation of a 2-dimensional sphere with a strict 4-coloring on its vertices can seen as the boundary of a triangulation of a 3-dimensional disk with the same vertices and preserving the 4-coloring.
Proof of existence for ideal triangulations that normalize fibers in certain 3-manifolds.
Efficient triangulations help in understanding 3-manifold boundaries.
We give a simple method to find ideal points of the character variety of a 3-manifold from an ideal triangulation.
This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program …
Infinite type surfaces can be perfectly divided into triangles.
Starting with an ideal triangulation of the interior of a compact 3-manifold M with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulations of M itself. Besides a step-by-step algorithm for such a construction,…
3-manifolds have covers with infinitely many ideal triangulations.
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…
Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.
Proves ideal triangulations of hyperbolic alternating links are non-degenerate.
Bounding shears in ideal triangulations on hyperbolic surfaces.
New isolated geometric triangulations found in once-punctured torus bundles.
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely man…
Establishes connection between Alexander polynomials and triangulations.
The paper proves ideal triangulations and disk unfolding for singular flat surfaces.
The paper studies polynomials and ideals from colored Jones polynomials for links.
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly -colored) triangulation of a combinatorial -manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following the…
Paper finds infinite family of minimal triangulations for complex 3D shapes.
New triangulations for twist knots, proving volume conjecture.
New method to parametrize infinite Riemann surfaces with bounded triangulations.
If all but two vertices of a triangulated sphere have degrees divisible by , then the exceptional vertices are not adjacent. This theorem is proved for with the help of the coloring monodromy. For colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can asso…
Study on non-peripheral ideal decompositions of alternating knots.
Refining the notion of an ideal triangulation of a compact three-manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,a), where M is a three-manifold and a is a collection of properly embedded arcs. We also show that certain well-understood combinatorial moves are sufficient to relate …
In previous joint work with Frohman and Lofaro a noncommutative generalization of the A-polynomial of a knot was introduced, consisting of a finitely generated ideal of polynomials (the noncommutative A-ideal) in the quantum plane. The present paper shows that the noncommutative A-ideal of a knot, together with finitel…
0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation or M can be shown to be one of the manifolds S^3, RP^3 or L(3,1). Similarly, any triangulation of a compact, orientable, …
Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a generalisation of the hyperbolic gluing equations, which enables the construction of hy…
It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a ne…
Decomposes skein algebras for surfaces.
New quantum invariant for framed 3-manifolds using ideal triangulations.
It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs us…
Ideal triangulations of 3-manifolds are shown equivalent up to certain moves.
Essential triangulations of certain manifolds are connected via specific moves.
Classifies positive integral friezes on surfaces.
We give a complete proof of Thurston's celebrated hyperbolic Dehn filling theorem, following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. Thi…
Proves hyperbolic 3-manifolds have angle structures under certain conditions.
Geodesic tetrahedra found for Platonic cusped manifolds.