Combinatorial Decompositions, Kirillov–Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra

We define invariants for colored oriented spatial graphs by generalizing CM invariants [F. Costantino and J. Murakami, On [Formula: see text] quantum [Formula: see text]-symbols and their relation to the hyperbolic volume, Quantum Topol. 4 (2013) 303–351], which were defined via non-integral highest weight representations of [Formula: see text]. We apply the same method used to define Yokota’s invariants, and we call these invariants Yokota type invariants. Then, we propose a volume conjecture of the Yokota type invariants of plane graphs, which relates to volumes of hyperbolic polyhedra corresponding to the graphs, and check it numerically for some square pyramids and pentagonal pyramids.

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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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On the volume and surface area of hyperbolic polyhedra

Geometriae Dedicata ◽

10.1007/bf00145916 ◽

1991 ◽

Vol 40 (2) ◽

Cited By ~ 2

Author(s):

Yosuke Miyamoto

Keyword(s):

Surface Area ◽

Hyperbolic Polyhedra

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Spacetime as a quantum circuit

Journal of High Energy Physics ◽

10.1007/jhep04(2021)207 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

A. Ramesh Chandra ◽

Jan de Boer ◽

Mario Flory ◽

Michal P. Heller ◽

Sergio Hörtner ◽

...

Keyword(s):

Path Integral ◽

Constant Scalar Curvature ◽

Quantum Circuit ◽

Quantum Circuits ◽

Special Role ◽

Gravitational Action ◽

Volume Conjecture ◽

Interesting Connection ◽

Kinematic Space ◽

Boundary States

Abstract We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the “complexity equals volume” conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $$ T\overline{T} $$ T T ¯ , we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action.

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ON THE VOLUME CONJECTURE FOR CLASSICAL SPIN NETWORKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511009522 ◽

2012 ◽

Vol 21 (03) ◽

pp. 1250022 ◽

Cited By ~ 8

Author(s):

ABDELMALEK ABDESSELAM

Keyword(s):

Invariant Theory ◽

Classical Literature ◽

Spin Networks ◽

Classical Invariant Theory ◽

Classical Spin ◽

Volume Conjecture ◽

Symbolic Method ◽

Exact Determination ◽

Classical Invariant

We prove an upper bound for the evaluation of all classical SU2 spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

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