Q-Series and quantum spin networks

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.

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q-DEFORMED SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216507005282 ◽

2007 ◽

Vol 16 (03) ◽

pp. 267-332 ◽

Cited By ~ 32

Author(s):

LOUIS H. KAUFFMAN ◽

SAMUEL J. LOMONACO

Keyword(s):

Quantum Algorithms ◽

Jones Polynomial ◽

Braid Groups ◽

Spin Networks ◽

Spin Network ◽

Topological Quantum ◽

Jones Polynomials ◽

Topological Quantum Computation ◽

Bracket Polynomial ◽

Colored Jones Polynomials

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

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AN EFFICIENT QUANTUM ALGORITHM FOR COLORED JONES POLYNOMIALS

International Journal of Quantum Information ◽

10.1142/s0219749908004092 ◽

2008 ◽

Vol 06 (supp01) ◽

pp. 773-778 ◽

Cited By ~ 1

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.

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On the 2-head of the colored Jones polynomial for pretzel knots

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haz021 ◽

2019 ◽

Vol 70 (4) ◽

pp. 1353-1370

Author(s):

Paul Beirne

Keyword(s):

Jones Polynomial ◽

Infinite Family ◽

Higher Order ◽

Careful Examination ◽

Colored Jones Polynomial ◽

Stability Properties ◽

Pretzel Knots

Abstract In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.

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Quantum modularity of partial theta series with periodic coefficients

Forum Mathematicum ◽

10.1515/forum-2020-0201 ◽

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) .

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Unzipping Natural Products: Improved Natural Product Structure Predictions by Ensemble Modeling and Fingerprint Matching

10.26434/chemrxiv.6863864.v1 ◽

2018 ◽

Author(s):

William A. Shirley ◽

Brian P. Kelley ◽

Yohann Potier ◽

John H. Koschwanez ◽

Robert Bruccoleri ◽

...

Keyword(s):

Natural Products ◽

Open Source ◽

Natural Product ◽

Gene Cluster ◽

Public Domain ◽

Product Structure ◽

Fingerprint Matching ◽

Ensemble Modeling ◽

Chemical Structures ◽

Source Gene

This pre-print explores ensemble modeling of natural product targets to match chemical structures to precursors found in large open-source gene cluster repository antiSMASH. Commentary on method, effectiveness, and limitations are enclosed. All structures are public domain molecules and have been reviewed for release.

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Faculty Opinions recommendation of Protein structure similarity clustering and natural product structure as guiding principles in drug discovery.

Faculty Opinions – Post-Publication Peer Review of the Biomedical Literature ◽

10.3410/f.1165740.628785 ◽

2009 ◽

Author(s):

Robert Powers

Keyword(s):

Protein Structure ◽

Drug Discovery ◽

Natural Product ◽

Product Structure ◽

Structure Similarity ◽

Guiding Principles

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Natural product fragment combination to performance-diverse pseudo-natural products

Nature Communications ◽

10.1038/s41467-021-22174-4 ◽

2021 ◽

Vol 12 (1) ◽

Cited By ~ 2

Author(s):

Michael Grigalunas ◽

Annina Burhop ◽

Sarah Zinken ◽

Axel Pahl ◽

José-Manuel Gally ◽

...

Keyword(s):

Natural Products ◽

Product Design ◽

Natural Product ◽

Chemical Space ◽

Biological Evaluation ◽

Product Structure ◽

Biologically Relevant ◽

Biologically Relevant Chemical Space

AbstractNatural product structure and fragment-based compound development inspire pseudo-natural product design through different combinations of a given natural product fragment set to compound classes expected to be chemically and biologically diverse. We describe the synthetic combination of the fragment-sized natural products quinine, quinidine, sinomenine, and griseofulvin with chromanone or indole-containing fragments to provide a 244-member pseudo-natural product collection. Cheminformatic analyses reveal that the resulting eight pseudo-natural product classes are chemically diverse and share both drug- and natural product-like properties. Unbiased biological evaluation by cell painting demonstrates that bioactivity of pseudo-natural products, guiding natural products, and fragments differ and that combination of different fragments dominates establishment of unique bioactivity. Identification of phenotypic fragment dominance enables design of compound classes with correctly predicted bioactivity. The results demonstrate that fusion of natural product fragments in different combinations and arrangements can provide chemically and biologically diverse pseudo-natural product classes for wider exploration of biologically relevant chemical space.

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THE COLORED JONES POLYNOMIALS OF DOUBLES OF KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006452 ◽

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.

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