arXiv research

Reidemeister's theorem proved using smooth functions and transversality.

New theorem for 4D links simplifies characterisation problem.

The abstract extends Reidemeister theorem to 3-manifolds using diagrams of links and bands.

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…

Proves a theorem for comparing surfaces in 3D space.

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…

Study of equivariant movie moves for involutive links.

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.

We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar theorem for split links, which provides a polynomial upper bound on the number of …

Anomaly term vanishes for smooth conical spaces, non-trivial for cones over tori.

We prove that the twisted Reidemeister torsion of a 3-manifold corresponding to a fibered class is monic and we show that it gives lower bounds on the Thurston norm. The former fixes a flawed proof in [FV10], the latter gives a quick alternative argument for the main theorem of [FK06].

Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.

This note is an exposition of Waldhausen's proof of Waldhausen's Theorem: the three-sphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister-Singer Theorem.

The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.

For a compact oriented 3-manifold with torus boundary the adjoint Reidemeister torsion is defined as a function on the $\mathrm{SL}_2(\mathbb{C})$-character variety depending on a choice of a boundary curve. Under reasonable assumptions, it is conjectured that the adjoint torsion satisfies a certain type of vanishing i…

Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.

Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was i…

A {\em good drawing\/} of $K_n$ is a drawing of the complete graph with $n$ vertices in the sphere such that: no two edges with a common end cross; no two edges cross more than once; and no three edges all cross at the same point. Gioan's Theorem asserts that any two good drawings of $K_n$ that have the same rotations …

Paper equates torsions on wedge singularities.

We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many cases.

In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion and demonstrate that Bismut-Zhang's equivariant Cheeger-Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and …

We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Mueller theorem on Ray-Singer torsion and Reidemeister torsion, to the case where the flat vector bundle over a closed manifold carries a nondegenerate symmetric bilinear form. As a consequence, we prove the Burghelea-Haller conjecture which gives an an…

Explicit formula for Reidemeister torsion of two-bridge knots.

Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.

Minimal sets of moves for isotopic knots and trivalent graphs identified.

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…

The paper defines functions from spherical curves and chord diagrams, proving invariance under specific Reidemeister moves.

In the present paper we give a simple proof of the fact that the set of virtual links with orientable atoms is closed. More precisely, the theorem states that if two virtual diagrams $K$ and $K'$ have orientable atoms and they are equivalent by Reidemeister moves, then there is a sequence of diagrams $K = K_1 \to...\to…

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.

Study counts sub-chord diagrams to classify spherical curves.

We study the Reidemeister torsion and the analytic torsion of the $m$ dimensional disc in the Euclidean $m$ dimensional space, using the base for the homology defined by Ray and Singer in \cite{RS}. We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms ar…

Minimal generating sets of Reidemeister moves identified and classified.

New sequences prove some link diagrams can't be transformed by specific moves.

Minimal sets of moves for rotational Reidemeister diagrams are identified.

Defines a new $SL_2(\mathbb{R})$-Casson invariant using Reidemeister torsions.

We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's th…

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.

Let X be a compact oriented Riemannian manifold and let $φ:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $φ$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $φ$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lin…

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.

Paper proves a generalized Torres formula for twisted Reidemeister torsion.

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 calculates surface torsion via pants decompositions.

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.

We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type-two cusps in a Rubinstein-…

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…