Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.09296

Augmented links, shadow links, and the TV volume conjecture: a geometric perspective

Dionne Ibarra·Emma N. McQuire·Jessica S. Purcell
View original source ↗Read source document ↗Export metadata ↓

Abstract

For hyperbolic 3-manifolds, the growth rate of their Turaev-Viro invariants, evaluated at a certain root of unity, is conjectured to give the hyperbolic volume of the manifold. This has been verified for a handful of examples and several infinite families of link complements, including fundamental shadow links. Fundamental shadow links lie in connected sums of copies of $S^1\times S^2$, and their complements are built of regular ideal octahedra. Another well-known family of links with complements built of regular ideal octahedra are the octahedral fully augmented links in the 3-sphere. The complements of these links are now known to be homeomorphic to complements of fundamental shadow links, using topological techniques. In this paper, we give a new, geometric proof that complements of octahedral fully augmented links are isometric to complements of fundamental shadow links. We then use skein theoretic techniques to determine formulae for coloured Jones polynomials of these links. In the case of no half-twists, this gives a new, more geometric verification of the Turaev-Viro volume conjecture for these links.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dionne Ibarra, Emma N. McQuire, Jessica S. Purcell. 2025-06-10. Augmented links, shadow links, and the TV volume conjecture: a geometric perspective. https://arxiv.org/abs/2506.09296

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

KEEP EXPLORING

Related papers

arXiv2026

Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

We build a large family of hyperbolic links with an explicit decomposition of the complement into right-angled hyperbolic polytopes of finite volume. Namely, in a series of papers A.D.Mednykn and A.Yu.Vesnin introduced a construction that for a given right-angled polytope $P$ in geometry $\mathbb L^3$, $\mathbb R^3$, $\mathbb S^3$, $\mathbb L^2\times \mathbb R$, $\mathbb S^2\times \mathbb R$ and a Hamiltonian cycle, theta-subgraph or $K_4$-subgraph $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ has a complete hyperbolic structure of finite volume and generalize this criterion to similar links in $3$-manifolds different from $S^3$. We prove that hyperbolic links $C_Γ$ are parametrized by nonselfcrossing Eulerian cycles, Eulerian theta-subgraphs and Eulerian $K_4$-subgraphs in hyperbolic right-angled $3$-polytopes of finite volume in $\mathbb L^3$ with $0$, $2$ or $4$ finite vertices. The complement $S^3\setminus C_Γ$ is glued of $4$, $8$ or $16$ copies of the corresponding right-angled polytope. We give a criterion when the link $C_Γ$ consists of mutually unlinked circles and prove that if such a link is nontrivial, then it contains the Borromean rings. The latter problem is motivated by the Efimov effect in quantum mechanics.

math.GT↗
arXiv2026

More Versions of Real Link Floer Homology

In this paper, we further develop the real link Floer homology defined by the first author. We introduce a new base-pointing convention that leads to a different version of real link Floer homology and show that this new theory is related to the old one by an exact triangle. We also define a real link Floer theory for multi-based strongly invertible links, which is a strong real Heegaard invariant, and take a first step toward a real link Floer TQFT. A computer implementation for the new theory via grid diagrams was written by Zhenkun Li. We also include an appendix containing real grid homology of more than 50 small knots.

math.GT↗
arXiv2026

Knots and the Sierpinski Tetrahedron

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.

math.GT↗