Issue 4, Volume 39, December 2014

2014

1. Trace as an Alternative Decategorification Functor Anna Beliakova , Zaur Guliyev , Kazuo Habiro , Aaron D. Lauda PDF

   ▸ Abstract

   Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called “decategorification” functor. Algebras are typically categorified to additive categories with additional structure, and decategorification is usually given by the (split) Grothendieck group. In this article, we study an alternative decategorification functor given by the trace or the zeroth Hochschild–Mitchell homology. We show that this form of decategorification endows any two representations of the categorified quantum $\mathfrak {sl}_{n}$ with an action of the current algebra $\mathbf {U}(\mathfrak {sl}_{n}[t])$ on its center.

   425-480
2. Non Semi-Simple ${\mathfrak {sl}(2)}$ Quantum Invariants, Spin Case Christian Blanchet , Francesco Costantino , Nathan Geer , Bertrand Patureau-Mirand PDF

   ▸ Abstract

   Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum ${\mathfrak {sl}(2)}$ were obtained by the last three authors in Costantino et al. (To appear in J. Topology. arXiv:1202.3553). In their construction, the quantum parameter $q$ is a root of unity of order $2r$ where $r > 1$ is odd or congruent to 2 modulo 4. In this paper, we consider the remaining cases where $r$ is congruent to zero modulo 4 and produce invariants of 3-manifolds with colored links, equipped with generalized spin structure. For a given 3-manifold $M$, the relevant generalized spin structures are (non canonically) parametrized by $H^{1}(M;\mathbb{C}/2\mathbb{Z})$.

   481-495
3. A Survey on the Turaev Genus of Knots Abhijit Champanerkar , Ilya Kofman PDF

   ▸ Abstract

   The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants.

   497-514
4. Deformations of nilHecke Algebras Benjamin Cooper PDF

   ▸ Abstract

   Presentations are given for the deformed nilHecke algebras which appear in the work of Bressler and Evens. The obstruction to the braid relation being satisfied by each algebra is given. A family of positively graded deformations of nilHecke algebras is defined and shown to be independent of the deformations determined by oriented cohomology theories.

   515-527
5. A Generating Series for Murakami-Ohtsuki-Yamada Graph Evaluations Stavros Garoufalidis , Roland van der Veen PDF

   ▸ Abstract

   Murakami-Ohtsuki-Yamada introduced an evaluation of certain oriented planar trivalent graphs with colored edges. This evaluation plays a key role in the evaluation of the colored HOMFLY polynomial of a link in 3-space and its Khovanov-Rozansky categorification. Our goal is to give a generating series formula for the evaluation of MOY graphs, which may be useful in categorification, and in the study of $q$-holonomicity of the colored HOMFLY polynomial.

   529-539
6. Recurrent Sequences of Polynomials in Three-Dimensional Topology Stavros Garoufalidis PDF

   ▸ Abstract

   A sequence of polynomials in several variables is recurrent if it satisfies a linear recursion with fixed polynomial coefficients. The Newton polytope of a recurrent sequence of polynomials is quasi-linear. Our goal is to give examples of recurrent sequences of polynomials that appear in three-dimensional topology, classical, and quantum.

   541-548
7. On the Degree of the Colored Jones Polynomial Efstratia Kalfagianni , Christine Ruey Shan Lee PDF

   ▸ Abstract

   The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As a result, we use colored Jones link polynomials to extract an invariant that detects semi-adequate links and discuss some applications.

   549-560
8. Euler’s Beta Function and Pentagon Relations Rinat Kashaev PDF

   ▸ Abstract

   The (quantum) pentagon relations satisfied by the 6j-symbols of representation categories underlay the existing constructions in quantum topology based on combinatorics of triangulations of three-dimensional pseudo manifolds. We discuss the recent result as reported by Kashaev et al. (2012) on an integral five-term relation satisfied by Euler’s beta function.

   561-566
9. Twisted Alexander Polynomials and Ideal Points Giving Seifert Surfaces Takahiro Kitayama PDF

   ▸ Abstract

   The coefficients of twisted Alexander polynomials of a knot induce regular functions of the $SL_{2}(\mathbb {C})$-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler–Shalen theory. It implies a partial affirmative answer to a conjecture by Dunfield, Friedl, and Jackson.

   567-574
10. Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks Toshitake Kohno PDF

    ▸ Abstract

    We describe a relationship between homological representations of the braid groups due to R. Lawrence, D. Krammer, and S. Bigelow and the monodromy representations of the Knizhnik-Zamolodchikov (KZ) connection. First, we describe the comparison theorem in the case of generic parameters. Then, we discuss the nongeneric case from the viewpoint of conformal field theory.

    575-598
11. Taut Foliations, Left-Orderability, and Cyclic Branched Covers Cameron Gordon , Tye Lidman PDF

    ▸ Abstract

    We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental groups, and are not L-spaces.

    599-635
12. Bounds of Hausdorff Measures of Tame Sets Ta Le Loi , Phan Phien PDF

    ▸ Abstract

    In this paper, we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, and so on. Moreover, we also give some explicit bounds for semi-algebraic or semi-Pfaffian cases, which depend only on the combinatoric data representing the objects involved.

    637-647
13. The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot Hitoshi Murakami PDF

    ▸ Abstract

    A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.

    649-710