A spin network generalization of the Jones Polynomial and Vassiliev invariants

R. Gambini; J. Griego; J. Pullin

doi:10.1016/s0370-2693(98)00182-8

Home/Archive/arXiv

arXiv · q-alg/9711014

A spin network generalization of the Jones Polynomial and Vassiliev invariants

R. Gambini·J. Griego·J. Pullin

View original source ↗Read source document ↗Export metadata ↓

Abstract

We apply the ideas of Alvarez and Labastida to the invariant of spin networks defined by Witten and Martin based on Chern-Simons theory. We show that it is possible to define ambient invariants of spin networks that (for the case of SU(2)) can be considered as extensions to spin networks of the Jones polynomial. Expansions of the coefficients of the polynomial yield primitive Vassiliev invariants. The resulting invariants are candidates for solutions of the Wheeler--DeWitt equations in the spin network representation of quantum gravity.

Explore related subjects

q-alg gr-qc hep-th math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

R. Gambini, J. Griego, J. Pullin. 1997-11-18. A spin network generalization of the Jones Polynomial and Vassiliev invariants. https://doi.org/10.1016/s0370-2693(98)00182-8

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2023

On Irreducibility of Tensor Products of Yangian Modules

We study the tensor product $V$ of any number of "elementary" irreducible modules over the Yangian of the general linear Lie algebra. An elementary module is determined by a skew Young diagram and by a complex parameter, and contains a vector called singular. We give sufficient conditions for cyclicity in $V$ of the tensor product of these singular vectors. By using this result, we give an irreducibility criterion for $V$ when each of the skew Young diagrams determining the tensor factors has rectangular shape.

q-alg↗

arXiv2022

On the parametrization of solutions of the Yang--Baxter equations

We study all five-, six-, and one eight-vertex type two-state solutions of the Yang-Baxter equations in the form $A_{12} B_{13} C_{23} = C_{23} B_{13} A_{12}$, and analyze the interplay of the `gauge' and `inversion' symmetries of these solution. Starting with algebraic solutions, whose parameters have no specific interpretation, and then using these symmetries we can construct a parametrization where we can identify global, color and spectral parameters. We show in particular how the distribution of these parameters may be changed by a change of gauge.

q-alg↗

arXiv2020

Higher-Dimensional Algebra I: Braided Monoidal 2-Categories

We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded in 4-dimensional space). Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the `center' of a semistrict monoidal category C. This is analogous to the construction of a braided monoidal category as the center, or `quantum double', of a monoidal category. As a corollary, our construction yields a strictification theorem for braided monoidal 2-categories.

q-alg↗