The paper defines normal forms for rational 3-tangles and shows a sequence of moves to transform one form to another.
problem Understanding and manipulating rational 3-tangles.
method Definition of normal forms and sequence of normal jump moves.
result There is a sequence of normal jump moves leading to equivalent normal forms of rational 3-tangles.
Study on 3D surfaces and tangles formed by Poncelet triangles.
problem Geometric and topological properties of 3D surfaces and tangles.
method Exploration of Poncelet triangles and associated points.
result Properties of 3D surfaces and tangles formed by Poncelet triangles.
Paper classifies rational 3-tangles using normal forms and minimal coordinates.
problem Classifying rational 3-tangles up to isotopy.
method Defined normal form and normal coordinate, investigated minimal coordinates, constructed contractible simplicial complex.
result Simplicial complex of normal forms is contractible, leading to classification of rational 3-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…
New tangles found for 21 ribbon knots with 10 or fewer crossings.
problem Characterizing ribbon knots through tangle forms.
method For each of 21 ribbon knots, found a 4-strand tangle that satisfies specific closure properties.
result Each of 21 ribbon knots can be represented by a tangle that meets the specified closure conditions.
In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tan…
Paper discusses gliding algorithm to transform tangle diagrams into a specific form.
problem Transforming tangle diagrams into a specific form.
method Gliding algorithm to bring tangle diagrams to Over-then-Under (OU) form.
result Obtained a braid classification result and extended it to virtual braids.
Decomposes prime alternating links into prime tangles.
problem Understanding how to decompose prime alternating links into prime tangles.
method Refined results from Menasco and Thistlethwaite, focusing on alternating property and pseudo-Montesinos links.
result If a prime alternating link can be decomposed into two prime tangles, it must be visible in an alternating link diagram or the link is pseudo-Montesinos.
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 R=\mathbb{Z} R = Z , realizability depends on divisibility of the alternating sum. For R = Z / p Z R=\mathbb{Z}/p\mathbb{Z} R = Z / p Z , all vectors are realizable. A 20-crossing tangle forms knots with unique polynomial properties.
problem Constructing knots with specific polynomial properties.
method Using a 20-crossing tangle T20 undetectable by Kauffman bracket modulo 2.
result Jones polynomial of knots formed equals 1 modulo 2r.
Simplified Khovanov complexes for rational tangles computed quickly.
problem Computing Khovanov complexes for rational tangles efficiently.
method Inductive algorithm to compute minimal complexes; matrix actions for subobject bigradings.
result Minimal complexes have simple backbone of non-zero morphisms and can be described by the reduced Burau representation.
The paper computes a knot's Kauffman bracket polynomial using recursive concatenation of a 4-tangle shadow.
problem Computing the Kauffman bracket polynomial for complex knots.
method Recursive concatenation of a 4-tangle shadow, followed by a closure operation and polynomial computation.
result A method to compute the Kauffman bracket polynomial for knots formed from 4-tangle shadows.
New method to compute tangle sums in character varieties.
problem Computing character varieties for tangle sums.
method Cut-and-paste methods to construct perturbed character varieties.
result Bounding cochains can be used to recover differential of I a t u r a l I^
atural I a t u r a l . Tangles improve clustering in various datasets.
problem Clustering diverse datasets efficiently and accurately.
method Tangles aggregate cuts to identify dense structures, leading to soft cluster characterization.
result Tangle framework generates hierarchical soft dendrograms for cluster exploration.
Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.
problem Calculating the Kauffman bracket polynomial for Celtic link shadows.
method Two complementary approaches: recursive relation and 4-tangle algebra.
result Derived Kauffman bracket polynomial for C K 4 2 n CK_4^{2n} C K 4 2 n shadows. A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to form the knot K. We call K a closure of G. An obstruction to embedding a genus-1 ta…
Researchers link tangle invariants for Khovanov and knot Floer homologies.
problem Relating tangle invariants for Khovanov and knot Floer homologies.
method Constructing algebraic DA bimodules for tangles and open braids, showing homotopy equivalence to Ozsvath-Szabo bimodules.
result Homotopy equivalence of DA bimodules for tangles and knot Floer homology.
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.
Study counts specific surfaces in Montesinos knots with 4 rational tangles.
problem Investigating closed essential surfaces in Montesinos knots with 4 rational tangles.
method Analyzing the number of closed, connected, essential, orientable surfaces of fixed genus in knot complements.
result Exactly 12 genus 2 surfaces and 8φ(g - 1) surfaces of genus greater than 2 are found, independent of knot crossings.
Study corrects previous work on knot Floer homology of certain pretzel knots.
problem Computing the Knot Floer Homology of specific pretzel knots.
method Applied peculiar modules theory for Floer homology of 4-ended tangles, using immersed curve interpretation.
result Corrected the rank of Knot Floer Homology for some pretzel knots.
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.
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.
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.
Introduces XC-tangles for quantum tangle invariants.
problem Quantum invariants of tangles and virtual tangles.
method Introduces XC-tangles and their equivalence to virtual upwards tangles.
result Extension of knot isotopy invariants to XC-tangles.
Study connects taffy pulling, fractions, and rational tangles.
problem Understanding the relationship between taffy pulling, fractions, and rational tangles.
method Developed a taffy analogue for Conway's characterization of rational tangles and gave a geometric connection.
result Direct geometric connection between rational tangles and taffy pulls.
Method for computing Khovanov homology of tangles.
problem Limited explicit computational studies of Khovanov homology for tangles.
method Arc reduction approach to compute Khovanov homology.
result Derived and computed Poincaré polynomials for simple and complex tangles.
Invariants for virtual rational tangles defined using Kauffman's bracket.
problem Defining invariants for virtual rational tangles.
method Recursive formula using Kauffman's bracket polynomial.
result Computed several examples of the invariant.
Paper explores conditions for constructing new quasi-alternating links.
problem Applying construction method to non-alternating and opposite type tangles.
method Substituting quasi-alternating crossings with rational tangles of same type.
result Identifies conditions and specific tangles for construction validity.
The paper extends knot contact homology to tangles and proves a gluing formula.
problem Defining and proving invariants for tangles in R 3 \mathbb R^3 R 3 . method Extending knot contact homology to tangles and proving a gluing formula.
result Tangle contact homology detects triviality of tangles.
New homotopy refinements for tangle invariants defined.
problem Stable homotopy refinements for tangle invariants.
method Defined stable homotopy refinements of Khovanov's arc algebras and tangle invariants.
result Stable homotopy refinements of Khovanov's arc algebras and tangle invariants defined.
Paper defines and analyzes mathematical equivalence of periodic tangles.
problem Understanding the equivalence of periodic tangles.
method Established mathematical framework, characterized isotopies, generalized results.
result Characterization of DP tangle equivalence based on motifs.
Updated polynomial for virtual tangles, compatible with decompositions.
problem Generalizing polynomial for virtual tangles.
method Updated multi-variable affine index polynomial, introduced Turaev moves.
result Polynomial compatible with tangle decompositions and crossing weight recovery.
Spatial graphs study tangle replacement with equivalence classes.
problem Differentiating spatial graphs and their properties.
method Tangle replacement on spatial graphs, focusing on handcuff graphs.
result One-to-one correspondence between neighborhood equivalence classes and tangles.
Paper defines linking numbers for periodic tangles.
problem Defining invariant linking numbers for periodic tangles.
method Defined linking numbers using motifs and showed invariance.
result Linking numbers are well-defined and invariant.
We show that for a tangle T T T with − ∂ 0 T ≅ ∂ 1 T -\partial^0T \cong \partial^1 T − ∂ 0 T ≅ ∂ 1 T the Hochschild homology of the tangle Floer homology C T ~ ( T ) \widetilde{\mathit{CT}}(T) CT ( T ) is equivalent to the link Floer homology of the closure T ′ = T / ( − ∂ 0 T ∼ ∂ 1 T ) T' = T/(-\partial^0T \sim \partial^1 T) T ′ = T / ( − ∂ 0 T ∼ ∂ 1 T ) of the tangle, linked with the tangle axis. In addition, we show that t…
Computes Kauffman bracket polynomial for specific 2-tangle shadows.
problem Calculating Kauffman bracket polynomial for complex tangle structures.
method Computed Kauffman bracket polynomial for specific 2-tangle shadows with up to 4 crossings.
result Computed polynomial for specific 2-tangle shadows.
New homotopy refinements for tangle invariants.
problem Stable homotopy refinements for tangle invariants.
method Refined Khovanov and Chen-Khovanov spectra.
result Induces refinements of platform algebras and invariants.
Solves tangle equations involving composite links under specific conditions.
problem Solving tangle equations with composite links and 2-bridge links.
method Uses algebraic and double branched cover properties to solve equations.
result Non-hyperbolic solutions to tangle equations involving composite links are found.
Generalized Wriggle polynomial for virtual tangles.
problem Defining a polynomial invariant for virtual tangles.
method Generalization of the Wriggle polynomial to virtual tangles, proving invariance.
result The generalized Wriggle polynomial is a Vassiliev invariant of order one for virtual knots.
The paper addresses the k k k -tangle enumeration problem. We introduce a notion of cascade diagram for k k k -tangle projections. An effective enumeration algorithm for projections is proposed based on cascade representation. Tangles projections with up to 12 crossings are tabulated. We provide also pictures of alternating …
We note that a rational 3 3 3 -tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational 3 3 3 -tangle diagrams up to isotopy. However, there is no perfect classification about rational 3 3 3 -tangle diagrams such as the classification of rational 2 2 2 -tangle diagrams cor…
A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangl…
New results on splitting tangles and spatial graphs.
problem Issues with previous splitting results about tangles and spatial graphs.
method Generalization of Menasco's result to other classes of links, tangles, and spatial graphs.
result New more general results for tangles and spatial graphs.
Proof of tangle theorem using graph forests.
problem Embedded tangles in links.
method Result about spanning forests for graphs.
result Short proof of Krebes's theorem.
Tangle machines are a topologically inspired diagrammatic formalism to describe information flow in networks. This paper begins with an expository account of tangle machines motivated by the problem of describing `covariance intersection' fusion of Gaussian estimators in networks. It then gives two examples in which ta…
New Alexander-type invariant for tangles defined and studied.
problem Generalization of Alexander polynomial to tangles.
method Definition of rMVA invariant from tMVA invariant, using Hodge star map and circuit algebras.
result rMVA invariant is a metamonoid morphism and equivalent to Bar-Natan's tangle invariant.