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 spherical curve deformations solve a conjecture.
Finite type invariants separate PL links in 3D space.
We construct real analytic flat Moebius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature.
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …
In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded -dimensional Lipschitz submanifolds in . It turns out that due to a smoothing effect any seq…
This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop fram…
Paper characterizes isotopies and hyperbolicity of weaves using geodesics.
New algebraic structure for 2-string links and long knots.
For Gamma a finite, connected metric graph, we consider the space of configurations of n points in Gamma with a restraint parameter r dictating the minimum distance allowed between each pair of points. These restricted configuration spaces come up naturally in topological robotics. In this paper, we study the homotopy,…
We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.
Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of…
Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band a…
Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result f…
Develops a framework for stuck knots with rigid constraints and invariants.
Introduces a Cost function to measure Legendrian knot obstructions.
Proves a theorem for comparing surfaces in 3D space.
New invariants define the rational and real homotopy types of closed manifolds.
Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on . Some of the knot types considered in this article provide new examples of non transversally simple knot types.
We prove the existence of Lagrangian fillings for -type Legendrian links.
Defines Lorentzian distance on contactomorphisms, proving continuity and finite conditions.
We prove a Chekanov-type theorem for the spherization of the cotangent bundle of a closed manifold . It claims that for Legendrian submanifolds in the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies.
New Legendrian knots found with equivalent Stein traces.
In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 s…
The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isoto…
New examples show transverse knots are determined by their branched covers.
A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transver…
We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the prob…
Study of cable links of uniformly thick knots, revealing new isotopy phenomena.
The Markov Theorem Without Stabilization (MTWS) (see math.GT/0310279) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive the…
We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF_4 by enumerating the connected components of the moduli space of real 2-elementary K3 surfaces of type (S,θ)=((3,1,1), -id). We also list up all the ca…
The paper studies knot types of clean intersections in a 3D space.
In this article we introduce a framization of the Hecke algebra of type B. For this framization we construct a faithful tensorial representation and two linear bases. We finally construct a Markov trace on these algebras and from this trace we derive isotopy invariants for framed and classical knots and links in the so…
We describe the first part of a gluing theory for the bigraded Khovanov homology with integer coefficients. This part associates a type D structure to a tangle properly embedded in a half-space and proves that the homotopy class of the type D structure is an invariant of the isotopy class of the tangle. The constructio…
Regular neighborhoods of singular submanifolds are isotopic to bundle morphisms.
Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). The construction of X^j(L) is combinatorial and explicit. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link.
We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.
A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.
New obstructions show some 4-manifold homeomorphisms are pseudo-isotopic but not isotopic.
We combine Freedman's topology with Eliashberg's holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded 2-complex K in a complex surface, after C^0-small topological ambient isotopy, is the int…
The paper proves the regularity of cohomogeneity two problems and constructs minimal hypersurfaces on spheres.
Approaches 4D Schoenflies via pseudo-isotopy.
We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a -manifold that are transverse to a nowhere-zero vector field up to the corresponding isotopy relation. Such knots are called …
New knot polynomials distinguish knot orientations without using knot groups.
Classifies curves up to symplectic isotopy.
In this paper we solve the following problem in the affirmative: Let be a continuum in the plane $\complex$ and suppose that $h:Z\times [0,1]\to\complex$ is an isotopy starting at the identity. Can be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar …
Smooth isotopy on cube saves energy with extra dimensions.
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In…
A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister II moves with opposite orientations and Reidemeister III moves with alternating orientations when checking the strands clockwise (or anticlockwise). We define a link polynomial derived from the Jones polynomial which is, in gen…