Recurrence Relations and Asymptotics of Colored Jones Polynomials

We consider $$q$$-difference equations for colored Jones polynomials. These sequences of polynomials are invariants for the knots and their asymptotics plays an important role in the famous volume conjecture for the complement of the knot to the $$3$$d sphere. We give an introduction to the theory of hyperbolic volume of the knots complements and study the asymptotics of the solutions of $$q$$-recurrence relations of high order.

Topological entanglement and hyperbolic volume

The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S3 complements of a two-component link which is a connected sum of a knot $$ \mathcal{K} $$ K and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ), which is the partition function of $$ {M}_{{\mathcal{K}}_m} $$ M K m . Here $$ {M}_{{\mathcal{K}}_m} $$ M K m is a closed 3-manifold associated with the knot $$ \mathcal{K} $$ K m, where $$ \mathcal{K} $$ K m is a connected sum of m-copies of $$ \mathcal{K} $$ K (i.e., $$ \mathcal{K} $$ K #$$ \mathcal{K} $$ K . . . #$$ \mathcal{K} $$ K ) which mimics the well-known replica method. We analayse the partition functions Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) can grow at most polynomially in k. On the contrary, we conjecture that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) is the hyperbolic volume of the knot complement S3\$$ \mathcal{K} $$ K m. We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.

The reduced Dijkgraaf–Witten invariant of double twist knots in the Bloch group of 𝔽p

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold [Formula: see text] can be obtained as the image of the Dijkgraaf–Witten invariant of [Formula: see text] by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and [Formula: see text] in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and [Formula: see text], and give a formula of them for [Formula: see text].

Topological Entropy of Pseudo-Anosov Maps from a Typical Thurston’s Construction

In this paper, we develop a way to extract information about a random walk associated with a typical Thurston’s construction. We first observe that a typical Thurston’s construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston’s construction that have finite 2nd moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane’s estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston’s construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston’s construction.

Disentangling a deep learned volume formula

We present a simple phenomenological formula which approximates the hyperbolic volume of a knot using only a single evaluation of its Jones polynomial at a root of unity. The average error is just 2.86% on the first 1.7 million knots, which represents a large improvement over previous formulas of this kind. To find the approximation formula, we use layer-wise relevance propagation to reverse engineer a black box neural network which achieves a similar average error for the same approximation task when trained on 10% of the total dataset. The particular roots of unity which appear in our analysis cannot be written as e2πi/(k+2) with integer k; therefore, the relevant Jones polynomial evaluations are not given by unknot-normalized expectation values of Wilson loop operators in conventional SU(2) Chern-Simons theory with level k. Instead, they correspond to an analytic continuation of such expectation values to fractional level. We briefly review the continuation procedure and comment on the presence of certain Lefschetz thimbles, to which our approximation formula is sensitive, in the analytically continued Chern-Simons integration cycle.

Borel and Volume Classes for Dense Representations of Discrete Groups

We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of a finite field

Let [Formula: see text] be a closed oriented 3-manifold and let [Formula: see text] be a discrete group. We consider a representation [Formula: see text]. For a 3-cocycle [Formula: see text], the Dijkgraaf–Witten invariant is given by [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], and [Formula: see text] denotes the fundamental class of [Formula: see text]. Note that [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], we consider an equivalent invariant [Formula: see text], and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of [Formula: see text] in terms of the image of the Dijkgraaf–Witten invariant for [Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text]. In this paper, by replacing [Formula: see text] with a finite field [Formula: see text], we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL[Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text].

The strong Suslin reciprocity law

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$ -theory. The Milnor $K$ -groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

Crosscap number of knots and volume bounds

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).