Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · q-alg/9601002

An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants

Thang T. Q. Le
View original source ↗Read source document ↗Export metadata ↓

Abstract

In [LMO] a 3-manifold invariant $Ω(M)$ is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant $Ω$ takes values in a graded Hopf algebra of Feynman 3-valent graphs. Here we show that for homology 3-spheres the invariant $Ω$ is {\em universal} for all finite type invariants, i.e. $Ω_n$ is an invariant of order $3n$ which dominates all other invariants of the same order. This shows that the set of finite type invariants of homology 3-spheres is equivalent to the Hopf algebra of Feynman 3-valent graphs. Some corollaries are discussed. A theory of groups of homology 3-spheres, similar to Gusarov's theory for knots, is presented.

Explore related subjects

q-algmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Thang T. Q. Le. 1996-01-16. An Invariant of Integral Homology 3-Spheres Which Is Universal For All Finite Type Invariants. https://arxiv.org/abs/q-alg/9601002

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

KEEP EXPLORING

Related papers

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↗
arXiv2019

Idempotents of Hecke algebras of type A

We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebras, namely the full curl and the sum of the Murphy operators. We discuss their calculation also in terms of the framing factor associated to the appropriate irreducible representation of the quantum group SU(N,q).

q-alg↗
arXiv2017

Dual Affine Quantum Groups

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.

q-alg↗