Reidemeister Torsion Research Articles

Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product Rep ( H ) ⋊ Aut ( H ) \text {Rep}(H)\rtimes \text {Aut}(H) . These are quantum invariants of knots endowed with a homomorphism of the knot group to Aut ( H ) \text {Aut}(H) . We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the S L ( n , C ) SL(n,\mathbb {C}) -twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant.

Adjoint Reidemeister torsions of some 3-manifolds obtained by Dehn surgeries

Abstract We determine the adjoint Reidemeister torsion of a $3$ -manifold obtained by some Dehn surgery along K, where K is either the figure-eight knot or the $5_2$ -knot. As in a vanishing conjecture (Benini et al. (2020, Journal of High Energy Physics 2020, 57), Gang et al. (2020, Journal of High Energy Physics 2020, 164), and Gang et al. (2021, Advances in Theoretical and Mathematical Physics 25, 1819–1845)), we consider a similar conjecture and show that the conjecture holds for the 3-manifold.

Adjoint twisted Reidemeister torsion and Gram matrices

We compute the adjoint twisted Reidemeister torsion for closed oriented hyperbolic 3-manifolds and for hyperbolic 3-manifolds with toroidal boundary. In our formula, we consider the manifold as obtained by doing a Dehn-filling along suitable boundary components of a fundamental shadow link complement, and the formula is in terms of the logarithmic holonomy of the meridians of the boundary components. As an important special case, we also write down a formula of the adjoint twisted Reidemeister torsion for the double of a hyperbolic 3-manifold with totally geodesic boundary in terms of the edge lengths of a geometric ideal triangulation of the manifold. These unexpected formulas were inspired by, and played an important role in, the study of the asymptotic expansion of quantum invariants [25].

The adjoint Reidemeister torsion for the connected sum of knots

Let K be the connected sum of knots K_1,\ldots,K_n . It is known that the \mathrm{SL}_2(\mathbb{C}) -character variety of the knot exterior of K has a component of dimension \geq 2 as the connected sum admits a so-called bending. We show that there is a natural way to define the adjoint Reidemeister torsion for such a high-dimensional component and prove that it is locally constant on a subset of the character variety where the trace of a meridian is constant. We also prove that the adjoint Reidemeister torsion of K satisfies the vanishing identity if each K_i does so.

A remark on Schottky representations and Reidemeister torsion

The present paper establishes a formula of Reidemeister torsion for Schottky representations. The theoretical results are applied to 3-manifolds with boundary consisting orientable surfaces with genus at least 2.

Enhanced Bruhat Decomposition and Morse Theory

AbstractMorse function is called strong if all its critical points have different critical values. Given such a function $f$ and a field $\mathbb{F}$ Barannikov constructed a pairing of some of the critical points of $f$, which is now also known as barcode. With every Barannikov pair (a.k.a. bar in the barcode), we naturally associate (up to sign) an element of $\mathbb{F} \setminus \{ 0 \}$; we call it Bruhat number. The paper is devoted to the study of these Bruhat numbers. We investigate several situations where the product of all the numbers (some being inversed) is independent of $f$ and interpret it as a Reidemeister torsion. We apply our results in the setting of one-parameter Morse theory by proving that generic path of functions must satisfy a certain equation mod 2 (this was initially proven in [ 2] under additional assumptions).On the linear-algebraic level, our constructions are served by the following variation of a classical Bruhat decomposition for $GL(\mathbb{F})$. A unitriangular matrix is an upper triangular one with 1s on the diagonal. Consider all rectangular matrices over $\mathbb{F}$ up to left and right multiplication by unitriangular ones. Enhanced Bruhat decomposition describes canonical representative in each equivalence class.

Quantum invariants of three-manifolds obtained by surgeries along torus knots

We study the asymptotic behavior of the Witten–Reshetikhin–Turaev invariant associated with the square of the n -th root of unity with odd n for a Seifert fibered space obtained by an integral Dehn surgery along a torus knot. We show that it can be described as a sum of the Chern–Simons invariants and the twisted Reidemeister torsions both associated with representations of the fundamental group to the two-dimensional complex special linear group.

From Three Dimensional Manifolds to Modular Tensor Categories

Using M-theory in physics, Cho et al. (JHEP 2020:115 (2020) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of $$\text {SL}(2,{\mathbb {C}})$$ -flat connections and $$\text {SL}(2,{\mathbb {C}})$$ -adjoint Reidemeister torsions of a three manifold can be packaged together to produce a $$(2+1)$$ -topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular T-matrix and the quantum dimensions of a candidate modular data. The modular S-matrix follows from essentially a trial-and-error procedure. We find premodular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our examples. Our main result is a mathematical construction of the modular data of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The modular data of premodular categories from Seifert fibered spaces can be realized using Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a $${\mathbb {Z}}_2$$ -homology sphere, and condensation of bosons in the resulting properly premodular categories leads to either modular or super-modular tensor categories.

Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.

An algebraic property of Reidemeister torsion

For a 3-manifold $M$ and an acyclic $\mathit{SL}(2,\mathbb{C})$-representation $\rho$ of its fundamental group, the $\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion $\tau_\rho(M) \in \mathbb{C}^\times$ is defined. If there are only finitely many conjugacy classes of irreducible representations, then the Reidemeister torsions are known to be algebraic numbers. Furthermore, we prove that the Reidemeister torsions are not only algebraic numbers but also algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3-manifolds. Also, for a knot exterior $E(K)$, we discuss the behavior of $\tau_\rho(E(K))$ when the restriction of $\rho$ to the boundary torus is fixed.

K-theoretic torsion and the zeta function

We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.

A Cheeger–Müller theorem for manifolds with wedge singularities

We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-M\"uller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.

A vanishing identity of adjoint Reidemeister torsions of twist knots

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 identities. In this paper, we prove that the conjecture holds for all hyperbolic twist knot exteriors by using Jacobi's residue theorem.

Holonomy invariants of links and nonabelian Reidemeister torsion

We show that the reduced \operatorname{SL}_2(\mathbb{C}) -twisted Burau representation can be obtained from the quantum group \mathcal{U}_q(\mathfrak{sl}_2) for q = i a fourth root of unity and that representations of \mathcal{U}_q(\mathfrak{sl}_2) satisfy a type of Schur–Weyl duality with the Burau representation. As a consequence, the \operatorname{SL}_2(\mathbb{C}) -twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet–Geer–Patureau-Mirand–Reshetikhin, and we interpret their invariant as a twisted Conway potential.

From torus bundles to particle–hole equivariantization

We continue the program of constructing (pre)modular tensor categories from 3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved is a collection of certain $\text{SL}(2, \mathbb{C})$ characters on a given manifold which serve as the simple object types in the corresponding category. Chern-Simons invariants and adjoint Reidemeister torsions play a key role in the construction, and they are related to topological twists and quantum dimensions, respectively, of simple objects. The modular $S$-matrix is computed from local operators and follows a trial-and-error procedure. It is currently unknown how to produce data beyond the modular $S$- and $T$-matrices. There are also a number of subtleties in the construction which remain to be solved. In this paper, we consider an infinite family of 3-manifolds, that is, torus bundles over the circle. We show that the modular data produced by such manifolds are realized by the $\mathbb{Z}_2$-equivariantization of certain pointed premodular categories. Here the equivariantization is performed for the $\mathbb{Z}_2$-action sending a simple (invertible) object to its inverse, also called the particle-hole symmetry. It is our hope that this extensive class of examples will shed light on how to improve the program to recover the full data of a premodular category.

Representation variety of free or surface groups and Reidemeister torsion

For $G \in \left\{ \mathrm{GL}(n,\mathbb{C}) , \mathrm{SL}(n,\mathbb{C})\right\} ,$ we consider $G-$valued representations of free or surface group with genus $ >1.$ We establish a formula for computing Reidemeister torsion of such representations in terms of Atiyah-Bott-Goldman symplectic form for $G.$ Furthermore, we apply the obtained results to hyperbolic 3-manifolds.

Infinitesimal gluing equations and the adjoint hyperbolic Reidemeister torsion

We establish a link between the derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. This provides a geometric reformulation of the non-abelian Reidemeister torsion corresponding to the adjoint of the monodromy representation of the hyperbolic structure. These results are then applied to the study of the `1-loop Conjecture' of Dimofte--Garoufalidis, which we generalize to arbitrary 1-cusped hyperbolic 3-manifolds. We verify the generalized conjecture in the case of the sister manifold of the figure-eight knot complement.

Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion

This paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at s=0 and its value at s=0 equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.

Twisted Reidemeister torsions via Heegaard splittings

We develop a new procedure for computing the twisted Reidemeister torsions of closed 3-manifolds with respect to SLn-representations. We further suggest three normalizations of the torsions and compare the normalized torsions to the preceding torsions. Moreover, we compute the twisted torsions of some Seifert 3-manifolds.

Non-semi-simple TQFT of the sphere with four punctures

In this work, we compute the representation of the mapping class group of the sphere with [Formula: see text] punctures arising from the non-semi-simple TQFT [C. Blanchet, F. Costantino, N. Geer and B. Patureau, Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants, Adv. Math. 301 (2016) 1–78]. We show that it is faithful. Lastly, we compare quantum representations of punctured spheres in general with those of braid groups.