Link invariants defined from finite crossed modules and Reidemeister pairs.
problem Defining link invariants from finite categorical groups.
method Definition of tangle and framed tangle invariants using finite crossed modules and Reidemeister pairs.
result Includes all rack and quandle cohomology (framed) link invariants and the Eisermann invariant of knots.
Formula calculates surface torsion via pants decompositions.
problem Computing Reidemeister torsion of compact surfaces.
method Used a specific pants decomposition to compute torsion.
result Formula for surface torsion in terms of pairs of pants.
We show that every knot type admits a pair of diagrams that cannot be made identical without using Reidemeister Omega_2-moves. We also show that our proof is compatible with known results for the other move types, in the sense that every knot type admits a pair of diagrams that cannot be made identical without using al…
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We…
New sequences prove some link diagrams can't be transformed by specific moves.
problem Proving some link diagrams can't be transformed by specific sequences of Reidemeister moves.
method Proved the existence of I-generalized ordered sequences to create simple transformations.
result Some link diagrams cannot be transformed by sequences of moves that increase, then decrease crossings.
New coloring method limits moves needed between virtual-link diagrams.
problem Determining the minimum moves needed between virtual-link diagrams.
method Introducing up-down colorings and using quandle cocycle invariants.
result Necessity of Reidemeister II moves for trivial virtual-knots.
Study of equivariant movie moves for involutive links.
problem Equivariant cobordisms between involutive links.
method Equivariant Morse theory and singularity theory.
result 39 equivariant movie moves for isotopic cobordisms.
Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure σon M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister-Turaev torsion of (M,σ), while the other one can be defined using the intersectio…
For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative tor…
New index principle shows indistinguishability of certain knot crossings.
problem Identifying indistinguishable crossings in knot diagrams.
method Developed a universal index function on knot diagrams that respects Reidemeister moves and sign of crossings.
result Crossings of the same sign in a classical knot diagram cannot be distinguished by any inherent property.
We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.
We give a construction to remove coincidence points of continuous maps on graphs (1-complexes) by changing the maps by homotopies. When the codomain is not homeomorphic to the circle, we show that any pair of maps can be changed by homotopies to be coincidence free. This means that there can be no nontrivial coincidenc…
If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by I. A. Dynnikov. Using this, Henrich and Kauffman gave an up…
The paper studies quandles of hyperbolic 3-space isometries.
problem Investigating quandles of hyperbolic 3-space isometries.
method Introducing a new quandle Q(Γ,γ) and constructing a canonical map to the conjugate quandle. result The canonical map from Q(Γ,γ) to the conjugate quandle is injective and has a discrete image. The paper shows that knot projections without triple chords can be simplified.
problem The study of knot projections and their chord diagrams.
method Flat Reidemeister moves that decrease 1-gons or strong 2-gons.
result For any knot projection without triple chords, a sequence of moves simplifies it to a simple closed curve.
Enhanced Bruhat decomposition studies Morse theory and Reidemeister torsion.
problem Investigating the Bruhat numbers associated with Morse functions.
method Using a variation of the classical Bruhat decomposition for GL(F). result The product of Bruhat numbers is independent of the Morse function and interpretable as Reidemeister torsion.
A linking pairing is a symetric bilinear pairing lambda: GxG --> Q/Z on a finite abelian group. The set of isomorphism classes of linking pairings is a non-cancellative monoid E under orthogonal sum, which is infinitely generated and infinitely related. We propose a new presentation of E that enables one to detect whet…
Explicit formula for Reidemeister torsion of two-bridge knots.
problem Calculating Reidemeister torsion for two-bridge knots.
method Provided an explicit formula and proved vanishing identities.
result Adjoint Reidemeister torsion satisfies vanishing identities.
Reidemeister's theorem proved using smooth functions and transversality.
problem Proving Reidemeister's theorem
method Using smooth functions and transversality
result Reidemeister's theorem proved
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
Study counts sub-chord diagrams to classify spherical curves.
problem Classifying spherical curves using chord diagrams.
method Counting sub-chord diagrams under specific moves.
result New invariant classifies prime reduced spherical curves.
Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
problem Recognizing and performing Reidemeister moves in Gauss diagrams.
method Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
result Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
The paper compares isotopic and diffeomorphic links in lens spaces.
problem Comparing isotopic and diffeomorphic links in lens spaces.
method Provides a set of moves on disk, band, and grid diagrams to connect diffeomorphic links.
result There are up to four isotopy equivalent links in each diffeo equivalence class.
The dual to a tetrahedron consists of a single vertex at which four edges and six faces are incident. Along each edge, three faces converge. A 2-foam is a compact topological space such that each point has a neighborhood homeomorphic to a neighborhood of that complex. Knotted foams in 4-dimensional space are to knotted…
Innovative rack theory applied to Legendrian links.
problem Classifying and distinguishing Legendrian links.
method Purely rack-theoretic approach, Legendrian Reidemeister moves, cusps, homogeneous representations, modules.
result Invariant distinguishes infinitely many Legendrian unknots and trefoils.
We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a lambda-regular SU(2) or SL(2, C)-representation of a knot group. Then we give a method to calculate the non-acyclic Reidemeister torsion of a knot exterior. We calculate a new example and investigate…
We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.
Minimal generating sets of Reidemeister moves identified and classified.
problem Classifying minimal generating sets of Reidemeister moves.
method Determined minimal generating sets, provided classifications, and identified candidates.
result 12 out of 16 candidates for minimal generating sets were proven minimal.
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
Defines a new SL2(R)-Casson invariant using Reidemeister torsions.
problem Computing and grading the SL2(R)-Casson invariant. method Using Reidemeister torsions to compute the invariant and discuss gradings.
result Defines a new invariant and provides methods for computation and grading.
Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians.
problem Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians. method Consider the sum of the adjoint Reidemeister torsions and prove integrality for twist knots and meridians.
result Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians. Calculates lower bounds for type III Reidemeister moves in link diagrams.
problem Determining the minimum number of Reidemeister III moves between link diagrams.
method Introducing non-self OU sequence and OU number for link diagrams.
result Provides a lower bound for the number of Reidemeister III moves.
We produce a facial state sum on plane diagrams of a knot or a link which admits an invariant specialization under Polyak's recent set of generating of 4 Reidemeister moves. Thus an isotopy invariant of framed links is obtained. Each state is a complete coloring of the faces of the diagram into white and black faces so…
Study higher dimensional Reidemeister torsion for twist knots surgeries.
problem Asymptotic behavior of Reidemeister torsion for exceptional surgeries.
method Analyzing graph manifolds and twist knot groups, determining limits of coefficients.
result Explicit set of limits for leading coefficients in higher dimensional Reidemeister torsion.
New theorem for 4D links simplifies characterisation problem.
problem Long-standing open problem in link characterisation.
method Reidemeister Theorem for solid ribbon torus links.
result Complete characterisation of a related class of links.
We provide an upper bound on the number of ordered Reidemeister moves required to pass between two diagrams of the same link. This bound is in terms of the number of unordered Reidemeister moves required.
Reidemeister torsion is algebraic for most 3-manifolds.
problem Characterizing algebraic properties of Reidemeister torsion.
method Defined algebraic numbers and proved them to be algebraic integers for specific 3-manifolds.
result Reidemeister torsions are algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3-manifolds.
Paper proves a generalized Torres formula for twisted Reidemeister torsion.
problem Alexander polynomial properties for links and sublinks.
method Uses twisted Reidemeister torsion to generalize Torres formula.
result Obtains a second proof of Morifuji's result.
In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and …
Formula connects analytic and Reidemeister torsions for hyperbolic manifolds.
problem Relating analytic and Reidemeister torsions for hyperbolic manifolds.
method Derives a formula relating analytic torsion to Reidemeister torsion of compactified manifolds.
result Formula connects analytic and Reidemeister torsions for hyperbolic manifolds.
Study calculates torsion for iterated torus knots using complex representations.
problem Calculating torsion for complex representations of iterated torus knots.
method Using complex special linear group representations of fundamental groups.
result Twisted Reidemeister torsions appear in colored Jones polynomial asymptotics.
Minimal set of directed Reidemeister moves identified for links.
problem Identifying the minimal set of directed Reidemeister moves for links.
method Proved the minimal generating set of 8 directed Polyak moves for directed oriented Reidemeister moves.
result Minimal generating set of 8 directed Polyak moves for directed oriented Reidemeister moves.
Researchers compute Reidemeister torsion for a specific 3-sphere.
problem Computing Reidemeister torsion for a specific type of 3-manifold.
method Numerical computations on representations of fundamental group in SL(2;C) and Reidemeister torsion.
result Corrected and presented new computations of Reidemeister torsion.
Study calculates Reidemeister torsions for 3-manifolds from specific surgeries.
problem Calculating Reidemeister torsions for 3-manifolds from Dehn surgeries.
method Determined adjoint Reidemeister torsion for specific 3-manifolds.
result Reidemeister torsion vanishes for the studied 3-manifolds.
In this note we present a short proof that the 4 oriented Reidemeister moves of type 2 together with any one of the 8 oriented Reidemeister moves of type 3 are sufficient to imply the other 7.
New physics-based method calculates Reidemeister torsions for hyperbolic 3-manifolds.
problem Calculating Reidemeister torsions for cusped hyperbolic 3-manifolds.
method Physics of wrapped M5-branes on the manifold.
result Rigorous proof and numerical verification for figure-eight knot complement and several knots.
Graphs from knot types help identify unique knots.
problem Identifying knots uniquely.
method Created Reidemeister graphs from knot types and analyzed their properties.
result Graph isomorphism type is a complete knot invariant.
As Oleg Viro describes in his paper, the most fundamental property of the Khovanov homology group is their invariance under Reidemeister moves. Viro constructes Khovanov complex and homology consisting of Jordan curves with sign and also gives a proof for the only case of first Reidemeister move by using his definition…