Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16731

On Intersection of Character Varieties Associated to Dehn Fillings of the Whitehead Link and Its Generalization

Jiayu Wan·Charles Katerba
View original source ↗Read source document ↗Export metadata ↓

Abstract

The canonical component of the character variety of a hyperbolic 2-bridge link complement M is an algebraic surface, from the work of Thurston. If we fix a boundary component T of M and perform a Dehn-filling on T, we get a 1-cusped manifold whose character variety is commonly a generalized curve that is naturally embedded in the character variety of M. If we perform distinct Dehn-fillings on M, we get various generalized curves on the canonical surface. In this paper, we analyze how these curves intersect with each other in the surface and study the topological implications for these intersection points. We also classify the intersection points that correspond to reducible characters, using the twisted Alexander polynomials associated to the Dehn-fillings. Throughout the paper, we use the Whitehead link as a motivating example to illustrate our core methodology, while some generalizations to other 2-bridge links will also be discussed.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jiayu Wan, Charles Katerba. 2026-10-12. On Intersection of Character Varieties Associated to Dehn Fillings of the Whitehead Link and Its Generalization. https://arxiv.org/abs/2610.16731

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

KEEP EXPLORING

Related papers

arXiv2026

Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets

Let $E_i$ be an oriented circle bundle over a closed oriented $n$-manifold $M_i$ with $i=1,2$ and assume that $M_2$ is aspherical. We prove the following geometric realization result, which extends the classical result for maps between circle bundles over surfaces to all dimensions $\geq 2$. Let $f\colon E_1\to E_2$ be a map. If the induced homomorphism $f_*$ on $π_1$ sends the fiber subgroup of $π_1(E_1)$ to the fiber subgroup of $π_1(E_2)$, then there is a fiber-preserving map in the homotopy class of $f$. Using the above result and a computation of the mapping degree set of fiber-preserving maps between $E_1$ and $E_2$, we obtain the following results on the finiteness and the realization problems of mapping degree sets. ($\mathcal F$) The mapping degree set $D(E_1, E_2)$ is finite if $M_2$ is hyperbolic and the Euler class of $E_2$ is not torsion. ($\mathcal R$) For any finite set $A$ of integers containing $0$ and any $n>2$, $A$ is the mapping degree set $D(M,N)$ for some closed oriented $n$-manifolds $M$ and $N$. Item ($\mathcal F$) extends the previously known $3$-dimensional case to all dimensions $\geq 3$. Item ($\mathcal R$) gives a complete answer to the realization problem on finite sets of integers (containing 0) in all dimensions, including in particular the previously unknown cases in dimensions $ 4$ and $5$.

math.GT↗
arXiv2026

Triple-cup product forms of 3-manifolds and Heegaard diagrams

The triple-cup product form $μ$ is a classical invariant of $3$-manifolds, determining the cohomology ring up to torsion. Given a closed, connected, oriented $3$-manifold $M$, we describe an explicit formula for computing $μ$ from a Heegaard diagram of $M$. Then, we show that the triple-cup product form $μ$ can be recovered as a reduction of Turaev's homotopy intersection form $η$ of the Heegaard surface.

math.GT↗
arXiv2026

A topological interpretation of numbers

We survey recent advancements on the realisation problem for mapping degree sets. In particular, we explain that any finite set containing zero is the mapping degree set between some $n$-manifolds, for each $n\geq3$. Our building factors for the construction of the realising manifolds will be aspherical manifolds. Thus, along the way, we review a couple of open questions related to maps of non-zero degree for aspherical manifolds.

math.GT↗