Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2108.02884

The skein algebra of the Borromean rings complement

Go Miura·Sakie Suzuki
View original source ↗Read source document ↗Export metadata ↓

Abstract

The skein algebra of an oriented $3$-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the $SL_2(\mathbb{C})$-character variety. In this paper we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in $S^3$.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Go Miura, Sakie Suzuki. 2023-01-09. The skein algebra of the Borromean rings complement. https://doi.org/10.1142/s0129167x22500495

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

KEEP EXPLORING

Related papers

arXiv2026

Splice formulae for Poincaré series of integral homology spheres

Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph $Γ$. One of its most important invariants is its multivariable Poincaré series (or zeta function) $f_Γ(\mathbf {t})$. Several numerical invariants can be read from $f_Γ(\mathbf{t})$, or even from its `polynomial part' $ \operatorname{Pol}_Γ(\mathbf{t})$. For example, the `normalized' Casson's invariant equals $ \operatorname{Pol}_Γ(1)$. In this note we provide several splice (surgery) formulae for $f_Γ$ and $\operatorname{Pol}_Γ$ (reduced to the node variables, or to the node variables of connected sub-graphs). In this way, these global invariants can be recovered from a collection of certain smaller graphs. Recall that some (integral homology sphere) 3-manifold invariants are additive with respect to the splice decomposition (like the Casson's invariant). However, some invariants need some `splice correction terms'. In our formulae the correction terms are easily computable one variable Alexander polynomials.

math.GT↗
arXiv2026

The Burau representation of the braid group is faithful for n = 4

In this paper we use ideas introduced earlier by Moody, Long, Long-Paton, and Bigelow to prove the theorem of the title, that the Burau representation of the classical braid group is faithful for n = 4. An immediate corollary is that the Jones representation of the braid group is also faithful for n = 4.

math.GT↗