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Andrew Yarmola

Publications and source records attributed to Andrew Yarmola.

10 recordsLinked to original sources
arXiv2024

Knots in circle bundles are determined by their complements

We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle $N$ over a genus $g \geq 2$ surface $S$ are determined by their complements. We apply this to the setting of canonical knots in the unit and projective tangent bundles, which are knots that are the set of tangents to a closed curve on $S$. We show that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of $S$.

math.GT↗
arXiv2022

On volumes and filling collections of multicurves

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $Γ$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hatΓ$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\hatΓ$ is the set of tangent lines to $Γ$. In particular, $vol(PT(S) \setminus \hatΓ)$ is a mapping class group invariant of the collection $Γ$. When $Γ$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil-Petersson distance between strata in Teichmüller space. Our main tool is the study of stratified hyperbolic links $\barΓ$ in a Seifert-fibered space $N$ over $S$. For such links, the volume of $N\setminus\barΓ$ is coarsely comparable to expressions involving distances in the pants graph.

math.GT↗
arXiv2021

Hyperbolic 3-manifolds of low cusp volume

We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings; to show that the figure-8 knot complement and its sister are the unique hyperbolic 3-manifolds with minimal volume maximal cusps; and to extend results on determining low volume closed and cusped hyperbolic 3-manifolds.

math.GT↗
arXiv2020

Random triangles on flat tori

Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this computation involves reducing the problem to new invariant of measurable sets in the plane that is unchanged under area-preserving affine transformations. Our result show that this probability is minimized at all rectangular tori and maximized at the regular hexagonal torus.

math.CO↗
arXiv2018

Properness for circle packings and Delaunay circle patterns on complex projective structures

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper.

math.GT↗
arXiv2018

The Two Eyes Lemma: a linking problem for horoball necklaces

In the course of our work on low-volume hyperbolic 3-manifolds, we came upon a linking problem for horoball necklaces in $\mathbb{H}^3$. A horoball necklace is a collection of sequentially tangent beards (i.e. spheres) with disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at most one and is tangent to the table. In this note, we analyze the possible configurations of an 8-bead necklace linking around two other diameter-one spheres on the table. We show that all the beads are forced to have diameter one, the two linked spheres are tangent, and that each bead must kiss (i.e. be tangent to) at least one of the two linked spheres. In fact, there is a 1-parameter family of distinct configurations.

math.GT↗
arXiv2017

Basmajian's identity in higher Teichmüller-Thurston theory

We prove an extension of Basmajian's identity to $n$-Hitchin representations of compact bordered surfaces. For $n=3$, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that, with respect to the Lebesgue measure on the Frenet curve associated to a Hitchin representation, the limit set of an incompressible subsurface of a closed surface has measure zero. This generalizes a classical result in hyperbolic geometry. Finally, we recall the Labourie-McShane extension of the McShane-Mirzakhani identity to Hitchin representations and note a close connection to Basmajian's identity in both the hyperbolic and the Hitchin settings.

math.GT↗
arXiv2016

The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary

In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic $n$-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold's volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, our extension adds terms corresponding to intrinsic invariants of boundary cusps.

math.GT↗
arXiv2015

An improved bound for Sullivan's convex hull theorem

Sullivan showed that there exists $K_0$ such that if $Ω\subset \hat{\mathbb{C}}$ is a simply connected hyperbolic domain, then there exists a conformally natural $K_0$-quasiconformal map from $Ω$ to the boundary ${\rm Dome}(Ω)$ of the convex hull of its complement which extends to the identity on $\partialΩ$. Explicit upper and lower bounds on $K_0$ were obtained by Epstein, Marden, Markovic and Bishop. We improve on these bounds, by showing that one may choose $K_0\le 7.1695$.

math.GT↗