Study on coloring virtual tangles with integer and modular arithmetic.
problem Characterizing Fox colorings of virtual tangle diagrams.
method Analyzed classical and virtual tangle diagrams using vector representations and divisibility conditions.
result For R=Z, realizability depends on divisibility of the alternating sum. For R=Z/pZ, all vectors are realizable. New methods found persistent tangles in knots.
problem Persistent tangles in knot diagrams.
method Non-trivial colorings for tangles.
result Any knot with non-trivial coloring has persistent tangles.
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…
This expository paper describes how the knot invariant Fox coloring can be applied to tangles.
Generalizes tangle Floer homology to A-tangles with A-colorings.
problem Extending tangle Floer homology to include A-colorings and cobordisms.
method Defined A-tangles, A-cobordisms, and tangle Floer homology functors for A-modules.
result Constructs tangle Floer homology for A-tangles and shows functoriality.
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 …
Functoriality proved for colored link invariants.
problem Link and tangle invariants functoriality proof.
method Functoriality proved for colored Khovanov-Rozansky invariants.
result Functoriality of colored link homologies proved.
Extended braid signature to colored tangles, linking to Maslov index and Turaev's functor.
problem Extending braid signature to colored tangles and understanding the homomorphism defect.
method Expressed the defect in terms of the Maslov index and Turaev's Lagrangian functor.
result Defect of multivariable signature expressed using Maslov index and Turaev's functor.
The paper examines when 2-string tangles can be embedded into specific link types.
problem When 2-string tangles can be embedded into the unknot, unlink, or split links.
method Geometric characterizations, tangle sums, and colorings.
result Prime 2-string tangles with up to seven crossings are classified for embedding into specific link types.
Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the de…
New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.
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…
Paper provides criteria to detect non-admissible quandles via coloring.
problem Determining non-admissibility of quandles.
method Using colorings of (1, 1)-tangles to detect non-admissibility.
result Constructed numerous non-admissible quandles.
The universal sl_2 invariant of bottom tangles has a universality property for the colored Jones polynomial of links. Habiro conjectured that the universal sl_2 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra U_h(sl_2) of the Li…
Study quantum groups and vertex algebras, linking tangle invariants and asymptotic dimensions.
problem Relationships between quantum groups and singlet vertex algebras.
method Use deformable families of modules to compute tangle invariants and relate to asymptotic dimensions.
result Regularized asymptotic dimensions of characters of singlet vertex algebras match modified traces of open Hopf link invariants.
New insights into Khovanov polynomials using tangle calculus.
problem Understanding the structure and evolution of Khovanov polynomials for long braids.
method Application of tangle calculus and evolution theory to Khovanov polynomials, focusing on jumps and thickness.
result Jumps in evolution are less frequent than expected, with most contributions being non-jumping.
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 …
Proves conjectures for pretzel knots using polynomial degrees.
problem Proving conjectures about pretzel knots.
method Using Hatcher-Oertel algorithm and colored Jones polynomial.
result Maximal degrees of colored Jones polynomial determine boundary slopes.
A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T, we prove that the universal invariant J_T of T associated to th…
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. Tangle functors link quantum representations to knot invariants.
problem Constructing tangle functors from semicyclic representations.
method Developed a tangle functor for framed homogeneous tangles colored with semicyclic representations.
result Invariant of (1,1)-tangles from knots matches Kashaev's invariant. A link L is called Brunnian if every proper sublink of L is trivial. Similarly, a bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we give a small subalgebra of the n-fold completed tensor power of U_h(sl_2) in which the universal sl_2 invariant of n-component Brunnian bottom…
New covering moves for 3-manifolds up to degree 4.
problem Relating colored link diagrams in 3-manifolds.
method Complete set of covering moves on braids in fixed degree d≥4. result Two local tangle replacements are sufficient after stabilization to the same degree at least 4.
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.
Categorifies quantum invariants using cobordism categories and operads.
problem Categorify quantum invariants using cobordism categories and operads.
method Constructs a cobordism category with a colored operad action, categorifies quantum sln invariants. result Consistency of the cobordism category and explicit functor to matrix factorizations conjectured.
Rational knots and links in solid torus characterized by continued fractions.
problem Characterizing rational knots and links in solid torus.
method Using rational tangles and continued fractions, and generalizing to skein module invariants.
result Rational links in solid torus fully characterized by rational tangles and continued fractions.
Researchers study rational and pretzel knots using affine group representations.
problem Understanding the structure and properties of rational and pretzel knots.
method Constructing representations of knot groups into the affine group AGL(1,ℂ) via a TQFT valued in spans of singular vector bundles.
result Closed-form expressions for Alexander polynomials and bounds on their zeros for rational and pretzel knots.
By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated…
We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then restrict our attention to plat closures of certain braids, a rich family of links who…
Abstract: New invariants for links and three-manifolds from finite groups.
problem Creating invariants for links and three-manifolds.
method Using finite groups to define R−matrices and extended R−matrices, constructing invariants through colored tangle categories. result Invariants for links and three-manifolds are group invariants.
Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.
problem Understanding Jones-Wenzl projectors in Khovanov spectra.
method Constructing and studying lifted projectors via maps and polynomial actions.
result Complete computation of 3-colored Khovanov spectrum of the unknot, proving conjectures.
Topologically protected vortex knots and links are proposed and proven.
problem Decaying of tangled vortex structures through local reconnections and strand crossings.
method Proposed and proven topologically protected vortex structures using non-Abelian topological vortices.
result Existence of topologically protected Q8-colored links and classification using the Q-invariant. Infinite group knot coloring polynomial generalizes quandle 2-cocycle invariant.
problem Generalizing knot invariants to infinite groups.
method Longitudinal mapping invariant based on meridian-longitude pair in knot group.
result Invariant values for specific knots and groups.
3D gauge theories link knot polynomials to vortex partition functions.
problem Connecting knot polynomials to gauge theory partition functions.
method Construct 3D N=2 abelian gauge theories on S2imesS1. result Colored Jones polynomials derived from vortex partition functions.
The slope of a colored link in an integral homology sphere is a rational function that generalizes the Kojima-Yamasaki η-function.
problem Defining and understanding the slope of colored links in integral homology spheres.
method Defining the slope as a rational function of Conway potentials and using generalized skein relations for tangles.
result The slope is responsible for an extra correction term in the signature formula for the splice of two links.
We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar-multiple of the sl(2) Reshetikhin-Turaev invariant. We expe…
New insights into non-torus links using topological vertices.
problem Understanding non-torus links through topological vertices.
method Tangle calculus and topological string considerations.
result Explicit description of a non-torus link L8n8. New knot invariants computed without explicit cocycles.
problem Computing knot invariants without explicitly finding cocycles.
method Using generalized Alexander quandles and colorings of 1-tangles.
result The 2-cocycle invariant distinguishes many prime knots.
Talk 1: Open problems in knot theory that everyone can try to solve. Knot theory is more than two hundred years old; the first scientists who considered knots as mathematical objects were A.Vandermonde (1771) and C.F.Gauss (1794). However, despite the impressive grow of the theory, there are simply formulated but funda…
The paper describes topological properties of arcs and crossings in knot theory.
problem Understanding the topological nature of arcs and crossings in knot theory.
method Topological description of arcs and crossings as isotopy classes of probes, homotopy classes of diagram elements.
result Sets of arcs and crossings are fundamental for algebraic objects like quandles, partial ternary quasigroups, biquandloids, and crossoids.
This paper is based on my talks (`Skein modules with a cubic skein relation: properties and speculations' and `Symplectic structure on colorings, Lagrangian tangles and its applications') given in Kyoto (RIMS), September 11 and September 18 respectively, 2001. The first three sections closely follow the talks: starting…
Enhanced trivalent tangles and handlebody-tangles invariants created.
problem Creating invariants for trivalent and handlebody-tangles.
method Using enhanced trivalent tangles and classical knot theory.
result Constructed invariants for trivalent and handlebody-tangles.
Classifies prime algebraic tangles up to 14 crossings.
problem Classifying prime algebraic tangles systematically.
method Developed a novel canonical representation to distinguish mutant tangles.
result Increased classification of prime tangles up to 14 crossings.
Paper defines new topological invariants for DP tangles.
problem Classifying and understanding doubly periodic tangles.
method Organized components into interlinked compounds; introduced axis-motif.
result Directional type is an invariant of DP tangles.
Simpler method detects trivial rational 3-tangle.
problem Detecting trivial rational 3-tangle.
method Bridge arc replacement method.
result Simpler method detects trivial rational 3-tangle.
Medial quandles fail to distinguish certain links, highlighting their limitations.
problem Detecting causality in spacetime using quandles.
method Investigated medial quandles' ability to distinguish specific links and knots.
result Medial quandles fail to distinguish certain links, including the connected sum of two Hopf links from an infinite series of relevant three-component links.