scholarly journals Quantum modularity of partial theta series with periodic coefficients

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ankush Goswami ◽  
Robert Osburn
Keyword(s):  
Modular Form ◽  
Jones Polynomial ◽  
Theta Series ◽  
Periodic Coefficients ◽  
Torus Knots ◽  
Colored Jones Polynomial ◽  
Root Of Unity ◽  
The Family

Abstract We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich–Zagier series F t ⁢ ( q ) \mathscr{F}_{t}(q) which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T ⁢ ( 3 , 2 t ) T(3,2^{t}) , t ≥ 2 t\geq 2 , is a weight 3 2 \frac{3}{2} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F ⁢ ( q ) F(q) .

scholarly journals OPTIMISTIC LIMITS OF THE COLORED JONES POLYNOMIALS AND THE COMPLEX VOLUMES OF HYPERBOLIC LINKS

2016 ◽  
Vol 100 (3) ◽  
pp. 303-337 ◽  
Author(s):  
JINSEOK CHO
Keyword(s):  
Classical Limit ◽  
Mathematical Formulation ◽  
Jones Polynomial ◽  
Physical Method ◽  
Saddle Point Method ◽  
Colored Jones Polynomial ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Colored Jones Polynomials ◽  
Actual Limit

The optimistic limit is a mathematical formulation of the classical limit, which is a physical method to estimate the actual limit by using the saddle-point method of a certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. This modified version is easier to handle and more combinatorial than the original one. On the other hand, it is known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at a certain root of unity. This optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it is very complicated and needs many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4{\it\pi}^{2}$. This new version is easier to handle and more combinatorial than the old version, and has many advantages over the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation.

scholarly journals The strong AJ conjecture for cables of torus knots

2015 ◽  
Vol 24 (14) ◽  
pp. 1550072 ◽  
Author(s):  
Anh T. Tran
Keyword(s):  
Jones Polynomial ◽  
Torus Knots ◽  
Colored Jones Polynomial ◽  
Pretzel Knots

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cable knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cable knots over torus knots.

scholarly journals AN EFFICIENT QUANTUM ALGORITHM FOR COLORED JONES POLYNOMIALS

2008 ◽  
Vol 06 (supp01) ◽  
pp. 773-778 ◽  
Author(s):  
MARIO RASETTI ◽  
SILVANO GARNERONE ◽  
ANNALISA MARZUOLI
Keyword(s):  
Jones Polynomial ◽  
Quantum Algorithm ◽  
Spin Network ◽  
Colored Jones Polynomial ◽  
Oriented Link ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

We construct a quantum algorithm to approximate efficiently the colored Jones polynomial of the plat presentation of any oriented link L at a fixed root of unity q. The construction exploits the q-deformed spin network as computational background. The complexity of such algorithm is bounded above linearly by the number of crossings of the link, and polynomially by the number of link strands.

scholarly journals Q-Series and quantum spin networks

2020 ◽  
pp. 1-19
Author(s):  
Mohamed Elhamdadi ◽  
Mustafa Hajij ◽  
Jesse S. F. Levitt
Keyword(s):  
Natural Product ◽  
Jones Polynomial ◽  
Infinite Family ◽  
Product Structure ◽  
Quantum Spin ◽  
Spin Networks ◽  
Spin Network ◽  
Colored Jones Polynomial ◽  
The Family

The tail of a quantum spin network in the two-sphere is a [Formula: see text]-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by [Formula: see text]. We compute the [Formula: see text]-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum spin networks under study satisfies a natural product structure.

THE COLORED JONES POLYNOMIALS OF DOUBLES OF KNOTS

2008 ◽  
Vol 17 (08) ◽  
pp. 925-937
Author(s):  
TOSHIFUMI TANAKA
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Skein Theory ◽  
Colored Jones Polynomials

We give formulas for the N-colored Jones polynomials of doubles of knots by using skein theory. As a corollary, we show that if the volume conjecture for untwisted positive (or negative) doubles of knots is true, then the colored Jones polynomial detects the unknot.

Nullification of Torus knots and links

2014 ◽  
Vol 23 (11) ◽  
pp. 1450058 ◽  
Author(s):  
Claus Ernst ◽  
Anthony Montemayor
Keyword(s):  
Torus Knots ◽  
The Family ◽  
Minimum Number ◽  
Knots And Links

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. In this paper we investigate the nullification numbers of a particular knot family, namely the family of torus knots and links.

scholarly journals Stability properties of the colored Jones polynomial

2019 ◽  
Vol 28 (08) ◽  
pp. 1950050
Author(s):  
Christine Ruey Shan Lee
Keyword(s):  
Power Series ◽  
Jones Polynomial ◽  
Topological Information ◽  
Colored Jones Polynomial ◽  
Stability Properties ◽  
Link Complement

It is known that the colored Jones polynomial of a [Formula: see text]-adequate link has a well-defined tail consisting of stable coefficients, and that the coefficients of the tail carry geometric and topological information on the [Formula: see text]-adequate link complement. We show that a power series similar to the tail of the colored Jones polynomial for [Formula: see text]-adequate links can be defined for all links, and that it is trivial if and only if the link is non [Formula: see text]-adequate.

ARITHMETIC ON A QUINTIC THREEFOLD

2001 ◽  
Vol 12 (08) ◽  
pp. 943-972 ◽  
Author(s):  
CATERINA CONSANI ◽  
JASPER SCHOLTEN
Keyword(s):  
Modular Form ◽  
Theta Series ◽  
Continuous System ◽  
Galois Representation ◽  
Hilbert Modular Form ◽  
Galois Action ◽  
Hilbert Modular ◽  
Common Eigenvector ◽  
Quintic Threefold ◽  
Real Quadratic

This paper investigates some aspects of the arithmetic of a quintic threefold in Pr 4 with double points singularities. Particular emphasis is given to the study of the L-function of the Galois action ρ on the middle ℓ-adic cohomology. The main result of the paper is the proof of the existence of a Hilbert modular form of weight (2, 4) and conductor 30, on the real quadratic field [Formula: see text], whose associated (continuous system of) Galois representation(s) appears to be the most likely candidate to induce the scalar extension [Formula: see text]. The Hilbert modular form is interpreted as a common eigenvector of the Brandt matrices which describe the action of the Hecke operators on a space of theta series associated to the norm form of a quaternion algebra over [Formula: see text] and a related Eichler order.

scholarly journals On Some Moves on Links and the Hopf Crossing Number

2020 ◽  
Vol 18 (1) ◽  
Author(s):  
Maciej Mroczkowski
Keyword(s):  
Primitive Root ◽  
Jones Polynomial ◽  
Crossing Number ◽  
Equivalence Classes ◽  
Lens Spaces ◽  
Root Of Unity ◽  
Arrow Diagrams

AbstractWe consider arrow diagrams of links in $$S^3$$ S 3 and define k-moves on such diagrams, for any $$k\in \mathbb {N}$$ k ∈ N . We study the equivalence classes of links in $$S^3$$ S 3 up to k-moves. For $$k=2$$ k = 2 , we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a k-th primitive root of unity is unchanged by a k-move, when k is odd. It is multiplied by $$-1$$ - 1 , when k is even. It follows that, for any $$k\ge 5$$ k ≥ 5 , there are infinitely many classes of knots modulo k-moves. We use these results to study the Hopf crossing number. In particular, we show that it is unbounded for some families of knots. We also interpret k-moves as some identifications between links in different lens spaces $$L_{p,1}$$ L p , 1 .

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