arXiv research

Categorifies colored Jones polynomial at roots of unity.

Researchers construct a series from knot data to match Kashaev invariant coefficients.

The paper proves a relation between four types of invariants.

Formula derived for special q-hypergeometric series at roots of unity.

For each finite dimensional, simple, complex Lie algebra $\mathfrak g$ and each root of unity $ξ$ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant $τ_M^{\mathfrak g}(ξ)\in \mathbb C$ of oriented 3-manifolds $M$. In the present paper we construct an invariant…

The study explores knot invariants using roots of unity.

A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $ωq$ where $ω$ is a complex root of unity, and under the substitution $q \to q^α$ where $α…

Study invariants for 3-manifolds at rational roots of unity.

Study on Jones polynomials and their roots in the unit circle and complex plane.

Jones polynomials have infinitely many roots of unity as zeros.

The paper extends ternary algebra concepts using cube roots of unity.

We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity ζgives the the SU(2) Witten-Reshetikhin-Turaev invariant τ_ζ(M) of M at ζ. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of…

For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.

Study centers of quantum tori and skein algebras for even roots of unity.

This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points whose associated roots of unity are not 1 or -1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than 1 and -1 occur.

Quantum group invariants from Lie superalgebra representations at roots of unity.

New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.

Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.

Constructs maps on skein modules using non-semisimple quantum invariants.

Study on quantum invariants of twist knots at specific roots of unity.

Hennings and Chern-Simons invariants match for certain quantum groups.

Study on quantum invariants of twist knots at specific roots of unity.

The Burau representation fails to be faithful at roots of unity for n ≥ 3.

New invariant derived from skein algebra representations at roots of unity.

The paper proves congruences for Fishburn numbers at roots of unity.

We consider subgroups of the braid groups which are generated by $k$-th powers of the standard generators and prove that any infinite intersection (with even $k$) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau's representations of the braid groups at roots of unity. Furtherm…

The paper studies the algebraic structure of Kauffman bracket skein at roots of unity.

Paper finds periodic patterns in colored Jones polynomial values.

We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $…

Quantum groups give lower genus bounds for links.

Researchers compute the rank and trace of Kauffman bracket skein algebra over its center.

Study shows link polynomial evaluations from Heegaard Floer theory.

The paper proves unicity for Kauffman bracket skein algebras.

We create Resthetikhin-Turaev topological invariants of closed orientable three-manifolds from the quantum supergroup U_q(osp(1|2n)) at certain even roots of unity. To construct the invariants we develop tensor product theorems for finite dimensional modules of U_q(osp(1|2n)) at roots of unity.

Hikami observed a discontinuity in a WRT invariant at roots of unity.

New idempotents defined in quantum groups at roots of unity, linking to SL2(Z) representations.

For a group G, the notion of a ribbon G-category was introduced by the second author in a previous work with a view towards constructing 3-dimensional homotopy quantum field theories (HQFT's) with target K(G,1). We discuss here how to derive ribbon G-categories from a simple complex Lie algebra g where G is the center …

New invariants explain topological properties of pseudo-Anosov maps.

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for appro…

Finite specializations of a q-deformed modular group at roots of unity.

We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $\exp({2π\sqrt{-1}}/{r})$ instead of the standard $\exp({π\sqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $\infty$, these invariants grow exponentially with growt…

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide examples of G-categories.

For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even …

Center identified in stated skein algebra for quantum traces.

We reprove and expand results of Bonahon and Wong on central elements of the Kauffman bracket skein modules at root of 1 and on the existence of the Chebyshev homomorphism, using elementary skein methods.

Researchers lift knot coloring polynomial to Habiro ring.

Study Type $C$ skein modules using $Sp(2n)$ webs and construct transparent elements.

Quantum $SL_2$ reveals surprising cancellations.