Issue 2, Volume 46, June 2021

2021

1. A Categorification of the Ribbon Element in Quantum $\mathfrak{sl}(2)$ Anna Beliakova , Kazuo Habiro PDF

   ▸ Abstract

   We define a bicomplex whose Euler characteristic is the idempotented version of the ribbon element of quantum $\mathfrak{sl}(2)$. We show that properties of this bicomplex descend to the centrality, invertibility, and symmetries of the ribbon element after decategorification.

   225-264
2. Stated Skein Modules of Marked 3-Manifolds/Surfaces, a Survey Thang T. Q. Lê , Tao Yu PDF

   ▸ Abstract

   We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces) and describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representation theory of the skein algebra. In particular, we show that the skein algebra of non-closed marked surface at any root of 1 is a maximal order. We give a full description of the Azumaya locus of the skein algebra of the puncture torus and give partial results for closed surfaces.

   265-287
3. Remarks on Jones Slopes and Surfaces of Knots Efstratia Kalfagianni PDF

   ▸ Abstract

   We show that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure 8 knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial

   289-299
4. Examples of Homological Torsion and Volume Growth Abhijit Champanerkar , Ilya Kofman PDF

   ▸ Abstract

   We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular planar lattice is given by the volume of a hyperbolic polyhedron.

   301-308
5. Random Walks on Graphs and Approximation of $L^2$-Invariants Andrew Kricker , Zenas Wong PDF

   ▸ Abstract

   In this work, we interpret right multiplication operators $R_{w}: l^{2}(G) \rightarrow l^{2}(G)$ as random walk operators on certain labelled graphs we employ that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. I. Grigorchuk and A. Żuk to these graphs gives a new interpretation and proof of a special case of W. Lück’s famous Theorem on the Approximation of $L^2$-Betti numbers for countable residually finite groups by means of exhausting towers of finite-index subgroups. In particular, using this interpretation, the theorem follows naturally from standard theorems in probability theory concerning the weak convergence of probability measures that are characterized by their moments. This paper is mainly a direct adaptation of the ideas of Grigorchuk, Zuk̇ and Lück to this setting. We aim to explain how these ideas are related and give a short exposition of them.

   309-319
6. Cancellations in the Degree of the Colored Jones Polynomial Christine Ruey Shan Lee , Roland van der Veen PDF

   ▸ Abstract

   We give an alternate expansion of the colored Jones polynomial of a pretzel link which recovers the degree formula in Garoufalidis et al. (Internat. J. Math.31(7):66, 2020). As an application, we determine the degrees of the colored Jones polynomials of a new family of 3-tangle pretzel knots.

   321-345
7. Projective Structures on a Hyperbolic 3-orbifold Joan Porti , Stephan Tillmann PDF

   ▸ Abstract

   We compute and analyse the moduli space of those real projective structures on a hyperbolic 3-orbifold that are modelled on a single ideal tetrahedron in projective space. Parameterisations are given in terms of classical invariants, traces, and geometric invariants, cross-ratios.

   347-355
8. The Strong AJ Conjecture for the Figure 8 Knot Hoang-An Nguyen , Anh T. Tran PDF

   ▸ Abstract

   Motivated by the theory of quantum A-ideals of Frohman-Gelca-LoFaro, the theory of q-holonomicity of quantum invariants of Garoufalidis-Le, and the AJ conjecture of Garoufalidis, Sikora formulated the strong AJ conjecture which relates the A-ideal and recurrence ideal of a knot in the 3-sphere. This conjecture has been verified for all torus knots and most of their cables. In this paper, we verify the strong AJ conjecture for the figure 8 knot.

   357-368
9. Twist Formulas for One-Row Colored $A_2$ Webs and $\mathfrak{sl}_3$ Tails of $(2, 2m)$-Torus Links Wataru Yuasa PDF

   ▸ Abstract

   <p>The $\mathfrak{sl}_3$ colored Jones polynomial $J_{\lambda}^{\mathfrak{s}\mathfrak{l}_{3}}(L)$ is obtained by coloring the link components with two-row Young diagram $\lambda$. Although it is difficult to compute $J_{\lambda}^{\mathfrak{s}\mathfrak{l}_{3}}(L)$ in general, we can calculate it by using Kuperberg $A_2$ skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in $A_2$ web space and compute $J_{(n,0)}^{\mathfrak{s}\mathfrak{l}_{3}}(T(2,2m))$ for an oriented $(2, 2m)$-torus link. These explicit formulas derives the $\mathfrak{sl}_3$ tail of $T(2, 2m)$. They also give explicit descriptions of the $\mathfrak{sl}_3$ false theta series with one-row coloring because the $\mathfrak{sl}_2$ tail of $T(2, 2m)$ is known as the false theta series.</p>
    

   369-387
10. Recent Progresses on the Volume Conjectures for Reshetikhin-Turaev and Turaev-Viro Invariants Tian Yang PDF

    ▸ Abstract

    In this survey, we give a review of the recent progresses on the Volume Conjectures for the Reshetikhin-Turaev and the Turaev-Viro Invariants, and on the related AMU Conjecture.

    389-398
11. Braid Group Action on Projective Quantum $\mathfrak{sl}(2)$ Modules Konstantinos Karvounis PDF

    ▸ Abstract

    We define a family of braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the center of the braid group has finite order on the specialized Lawrence representations, this action is faithful for our extension.

    399-440