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.
Classifies prime algebraic tangles up to 14 crossings.
Extends tangle theory to include undetermined crossings in periodic structures.
We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant und…
Operator on tangles derived from knot 2-cabling.
An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for knotted trivalent graphs. Our method also yields invariants of, what we refer to as…
We compute Khovanov homology for tangles using TQFT.
Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots, and flat virtual knots…
We show how the theory of tangles is equivalent to that of well-connected tangles. These are drawn on a surface with boundary, and equivalent via Reidemeister moves of a restricted kind. This reworking of the graphical foundations for link and tangle theory can be expected to have a variety of applications, including o…
Paper defines and analyzes mathematical equivalence of periodic tangles.
The paper extends knot contact homology to tangles and proves a gluing formula.
We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the …
The central discovery of conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire huge Hilbert space of states. Somewhat similar, when a link diagram is glued from t…
New results on splitting tangles and spatial graphs.
Diagrammatic method calculates knot invariant from tangle decompositions.
We construct a bigraded (co)homology theory which depends on a parameter a, and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl(3) theory for foams on the other side. Our theory is properly functorial under tangle cobordisms…
Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted subm…
Introduces XC-tangles for quantum tangle invariants.
Tangles improve clustering in various datasets.
L-space knots lack essential Conway spheres, proven with Floer theory.
New method for knot closures from 1-tangles and annulus twists.
Extended symmetric union with multiple tangle regions and Alexander polynomial properties.
We discuss several open problems in classical Knot Theory and we develop techniques that allow us to study them: Lagrangian tangles, skein modules and Burnside groups.
Refined 1-cocycle for knots helps quantify isotopies.
Study tangle equations linking enzyme actions to knot theory.
New methods use Conway tangles to generate knots and links.
Refines a tangle invariant using XC-algebras.
Khovanov homology for pro-tangles and spectral sequences
A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of…
New method for simplifying knots with specific properties.
Categorifies Jones polynomial using Lie theory.
Alternative proof of Khovanov's up-to-sign functoriality for odd Khovanov homology.
We reprove and extend a result of David Krebes (J. Knot Theory Ramif. 8 (1999), 321-352) giving an obstruction to embedding a tangle T into a link L. Closing the tangle up in the two obvious ways gives rise to two links, the numerator and denominator links n(T) and d(T). Applying a homological argument to the 2-fold br…
With a 4-ended tangle , we associate a Heegaard Floer invariant , the peculiar module of . Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which recovers link Floer homology . Moreover, we classify…
Quantum theory constructs a group and skein module for knot complements.
Quantizes Chern-Simons invariant for tangle exteriors.
New Alexander polynomial for singular knots improves upon existing methods.
Paper provides criteria to detect non-admissible quandles via coloring.
The paper introduces a cobordism for Khovanov homology crossing change and categorifies Vassiliev skein relations.
We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress t…
Proves a conjecture about concordance invariant simplifying its relation to Rasmussen's invariant.
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…
Paper introduces untangling number to quantify 3-periodic tangle complexity.
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…
We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mut…
Paper defines new topological invariants for DP tangles.
Simpler method detects trivial rational 3-tangle.
This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox 3-coloring. We show also a relation between 3-colorings and the Jones polynomial. In …
Defines odd Khovanov homology via categorification of q-Schur algebra.