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.
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The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.
problem Existence of Poincaré embeddings for specific spaces.
method Relates total obstruction to Reidemeister trace and uses Poincaré duality.
result Diagonal maps admit Poincaré embeddings under certain conditions.
We give a homotopy invariant construction of the Reidemeister trace for the coincidence of two maps between closed manifolds of not necessarily the same dimensions. It is realized as a homology class of the homotopy equalizer, which coincides with the Hurewicz image of Koschorke's stabilized bordism invariant. To defin…
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 …
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…
The paper studies obstructions to homotopy invariance of loop coproducts.
problem Characterizing obstructions to homotopy invariance of loop coproducts.
method Using a construction of Geoghegan and Nicas, the paper defines the Reidemeister trace and realizes the Goresky-Hingston coproduct as a map of spectra.
result The failure of a map to entwine spectral coproducts can be characterized by Chas-Sullivan multiplication with the Reidemeister trace.
The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.
problem Defining and analyzing the adjoint Reidemeister torsion for connected sums of knots.
method Defined a natural way to compute the adjoint Reidemeister torsion for high-dimensional components of the character variety.
result The adjoint Reidemeister torsion is locally constant and satisfies the vanishing identity.
Researchers compute torsion for homology cylinders, proving it a finite-type invariant.
problem Computing finite-type invariants for homology cylinders.
method Non-commutative Reidemeister-Turaev torsion, using -group and group ring.
result The torsion's reduction is a finite-type invariant of degree .
If is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation . Porti defined a non-abelian Reidemeister torsion in a neighborhood of $ρ…
Let M be a closed 3-manifold obtained by Dehn surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any irreducible SL(2;C)-representation. It is described as a rational expression of the trace of the image of the meridian.
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
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.
We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group a…
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.
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.
Paper constructs a HOMFLYPT-type invariant for pseudo links.
problem Inability to construct polynomial invariants for pseudo links using Hecke algebra techniques.
method Using a resolution homomorphism and pseudo Hecke algebra of type \(A\), the paper constructs a HOMFLYPT-type invariant for oriented pseudo links.
result The constructed invariant satisfies a natural pseudo skein relation and admits a state-sum formulation.
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…
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.
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.
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.
Introduces a new triple coproduct for knots on surfaces, preserving local crossing patterns.
problem Tackles the lack of fine-grained detection in classical cobrackets for local crossing patterns.
method Defines an integer-valued invariant using a coproduct and intersection theory, extending Turaev's cobracket theory.
result Reveals an intrinsic simplicity in the algebraic framework, uniquely determining relations in the word space.
Defines a new -Casson invariant using Reidemeister torsions.
problem Computing and grading the -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- indices with adjoint Reidemeister torsions for twist knots and meridians.
problem Prove integrality of genus- 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- 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.
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.
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.
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.
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.
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…
New nonacyclic Reidemeister torsions defined for odd-dimensional manifolds.
problem Defining torsions for non-acyclic cohomology in odd-dimensional manifolds.
method Introduced a new Reidemeister torsion for manifolds of odd dimension, even when cohomology is not acyclic.
result Introduced new topological invariants and computed Reidemeister torsions for specific manifolds.
Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type IImath.GT
We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of a quandle cocycle invariant, we determine the necessity of Reidemeister moves of…
Paper shows unique decomposition of 3-manifolds and multiplicative property of Reidemeister torsion.
problem Unique decomposition of compact 3-manifolds.
method Study of adjoint Reidemeister torsion on disk sum decompositions.
result Adjoint Reidemeister torsion has a multiplicative property on unique decompositions.
The gluing equations of a cusped hyperbolic 3-manifold are a system of polynomial equations in the shapes of an ideal triangulation $\calT$ of that describe the complete hyperbolic structure of and its deformations. Given a Neumann-Zagier datum (comprising the shapes together with the gluing equations in a …
The paper consists of four parts. Part I presents a brief survey of the Nielsen fixed point theory. Part II deals with dynamical zeta functions connected with Nielsen fixed point theory. Part III is concerned with congruences for the Reidemeister and Nielsen numbers. Part IV deals with the Reidemeister torsion . In Cha…
These notes accompany some lectures given at the autumn school "Tresses in Pau" in October 2009. The abelian Reidemeister torsion for 3-manifolds, and its refinements by Turaev, are introduced. Some applications, including relations between the Reidemeister torsion and other classical invariants, are surveyed.
For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller the…
The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.
problem Understanding knot types in bounded ropelength sublevel spaces.
method Study thick representatives in bounded ropelength sublevel spaces through lifted Reidemeister graphs.
result Define characteristic Reidemeister patterns and finite recognition length.
We use higher parallel transport -- more precisely, the integration A_{infty}-functor constructed by Block-Smith and Arias Abad-Schaetz -- to define Reidemeister torsion for flat superconnections. We hope that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu, thus perm…
In this article, we give an explicit formula to compute the non-abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.
Virtual Knot Groupsmath.GT
New groups defined from knot diagrams, invariant under Reidemeister moves.
problem Classical knot groups are not invariant under all Reidemeister moves.
method Define quotient groups based on knot diagrams, invariant under Reidemeister moves.
result New groups include extended knot groups and are invariant under all Reidemeister moves.
Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being unknotted with resp…
We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expa…
We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PS…
Paper solves the minimal generating set problem for singular Reidemeister moves.
problem Determine minimal generating sets of oriented singular Reidemeister moves.
method Introduced new invariant for singular links to detect type IV moves and provide obstructions.
result Proved exactly 96 distinct inclusion-minimal generating sets for singular moves.
Recently twisted and higher order Alexander polynomials were used by Cochran, Harvey, Friedl--Kim and Turaev to give lower bounds on the Thurston norm. We first show how Reidemeister torsion relates to these Alexander polynomials. We then give lower bounds on the Thurston norm in terms of the Reidemeister torsion which…