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Yo'av Rieck

Publications and source records attributed to Yo'av Rieck.

At least 19 recordsLinked to original sources
arXiv2022

A Structure Theorem for Bad 3-Orbifolds

We explicitly construct a collection of bad 3-orbifolds, \(\mathcal{X}\), satisfying the following properties: \begin{enumerate} \item The underlying topological space of any \(X \in \mathcal{X}\) is homeomorphic to $S^2\times I$ or $(S^2\times S^1)\backslash B^3$. \item The boundary of any \(X \in \mathcal{X}\) consists of one or two spherical 2-orbifolds. \item Any bad 3-orbifold is obtained from a good 3-orbifold by repeating, finitely many times, the following operation: remove one or two orbifold-balls, and glue in some \(X \in \mathcal{X}\). \end{enumerate} Conversely, any bad 3-orbifold \(\OO\) contains some \(X \in \mathcal{X}\) as a sub-orbifold; we call removing \(X\) and capping the resulting boundary \em cut-and-cap.\em\ Then by cutting-and-capping finitely many times we obtain a good orbifold.

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arXiv2022

Strongly aperiodic SFTs on hyperbolic groups: where to find them and why we love them

D. B. Cohen, C. Goodman-Strauss, and the author proved that a hyperbolic group admits an "SA SFT" if and only if it has at most one end. This paper has two distinct parts: the first is a conversation explaining what an SA SFT is and how they may be of use. In the second part I attempt to explain both old and new ideas that go into the proof. References to specific claims in the original paper are given, with the hope that any interested reader may be able to find the details there more accessible after reading this exposition.

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arXiv2020

Thermodynamic metrics on outer space

In this paper we consider two piecewise Riemannian metrics defined on the Culler-Vogtmann outer space which we call the entropy metric and the pressure metric. As a result of work of McMullen, these metrics can be seen as analogs of the Weil-Petersson metric on the Teichmüller space of a closed surface. We show that while the geometric analysis of these metrics is similar to that of the Weil-Petersson metric, from the point of view of geometric group theory, these metrics behave very differently to the Weil-Petersson metric. Specifically, we show that when the rank $r$ is at least 4, the action of ${\rm Out}(\mathbb{F}_r)$ on the completion of the Culler-Vogtmann outer space using the entropy metric has a fixed point. A similar statement also holds for the pressure metric.

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arXiv2018

The unbearable hardness of unknotting

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the $4$-ball Euler characteristic, the slicing number, and the $4$-dimensional clasp number).

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arXiv2018

Embeddability in $\mathbb{R}^3$ is NP-hard

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases which can be solved in linear time,and the latter with a variety of computational problems in 3-manifold topology (for example, unknot or 3-sphere recognition, which are in NP and co-NP assuming the Generalized Riemann Hypothesis). Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings on link complements.

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arXiv2015

The spectrum of the growth rate of the tunnel number is infinite

In a previous paper Kobayashi and Rieck defined the growth rate of the tunnel number of a knot $K$, a knot invariant that measures the asymptotic behavior of the tunnel number under iterated connected sum of $K$. We denote the growth rate by $\mbox{gr}_t(K)$. In this paper we construct, for any $ε> 0$, a hyperbolic knots $K \subset S^{3}$ for which $1 - ε< \mbox{gr}_t(K) < 1$. This is the first proof that the spectrum of the growth rate of the tunnel number is infinite.

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arXiv2015

The growth rate of the tunnel number of m-small knots

In a previous paper the authors defined the growth rate of the tunnel number of knots, an invariant that measures that asymptotic behavior of the tunnel number under connected sum. In this paper we calculate the growth rate of the tunnel number of m-small knots in terms of their bridge indices.

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arXiv2014

Strong cylindricality and the monodromy of bundles

A surface $F$ in a 3-manifold $M$ is called cylindrical if $M$ cut open along $F$ admits an essential annulus $A$. If, in addition, $(A, \partial A)$ is embedded in $(M, F)$, then we say that $F$ is strongly cylindrical. Let $M$ be a connected 3-manifold that admits a triangulation using $t$ tetrahedra and $F$ a two-sided connected essential closed surface of genus $g(F)$. We show that if $g(F)$ is at least $38 t$, then $F$ is strongly cylindrical. As a corollary, we give an alternative proof of the assertion that every closed hyperbolic 3-manifold admits only finitely many fibrations over the circle with connected fiber whose translation distance is not one, which was originally proved by Saul Schleimer.

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arXiv2013

Hyperbolic volume and Heegaard distance

We prove (Theorem~1.5) that there exists a constant $Λ> 0$ so that if $M$ is a $(μ,d)$-generic complete hyperbolic 3-manifold of volume $\vol[M] < \infty$ and $Σ\subset M$ is a Heegaard surface of genus $g(Σ) > Λ\vol[M]$, then $d(Σ) \leq 2$, where $d(Σ)$ denotes the distance of $Σ$ as defined by Hempel. The key for the proof of the main result is Theorem~1.8 which is on independent interest. There we prove that if $M$ is a compact 3-manifold that can be triangulated using at most $t$ tetrahedra (possibly with missing or truncated vertices), and $Σ$ is a Heegaard surface for $M$ with $g(Σ) \geq 76t+26$, then $d(Σ) \leq 2$.

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arXiv2012

The Link Volume of Hyperbolic 3-Manifolds

We prove that for any V>0, there exist a hyperbolic manifold M_V, so that Vol(M_V) < 2.03 and LinVol(M_V) > V. The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain: 1) Let K be a component of a link L in S^3. Then "most" slopes on K cannot be completed to a cosmetic surgery on L, unless K becomes a component of a Hopf link. 2) Let X be a manifold and ε>0. Then all but finitely many hyperbolic manifolds obtained by filling X admit a geodesic shorter than ε (note that this finite set may correspond to an infinitely many fillings).

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arXiv2012

The Link Volumes of some prism manifolds

In arxiv:1205.1274 Rieck and Yamashita defined the link volume of 3-manifolds and studied some of its basic properties. Many of these properties are similar to the corresponding properties of the hyperbolic volume. In this paper we calculate the link volume of an infinite family of prism manifolds. As a corollary, we show that (in contrast to the hyperbolic volume) the link volume is not finite-to-one.

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arXiv2012

A linear bound on the tetrahedral number of manifolds of bounded volume (after Jorgensen and Thurston)

We provide a detailed proof of the following folklore theorem: Let mu > 0 be a Margulis constant for 3-dimensional hyperbolic space. Then for any d>0 there exists a constant K>0, depending on mu and d, so that for any complete finite volume hyperbolic 3-manifold M, the d-neighborhood of the mu-thick part of M can be triangulated using at most K Vol(M) tetrahedra; here Vol is the hyperbolic volume function. As a corollary, we obtain the following topological interpretation of the volume: the minimal number of tetrahedra required to triangulate a link exterior in M is linearly equivalent to Vol(M); for a precise statement see Corollary 1.3.

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arXiv2012

The Link Volume of 3-Manifolds

We view closed orientable 3-manifolds as covers of S^3 branched over hyperbolic links. For a p-fold cover M \to S^3, branched over a hyperbolic link L, we assign the complexity p Vol(S^3 minus L) (where Vol is the hyperbolic volume). We define an invariant of 3-manifolds, called the link volume and denoted LV, that assigns to a 3-manifold M the infimum of the complexities of all possible covers M \to S^3, where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently M can be represented as a cover of S^3. We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold M, Vol(M) < LV(M). We prove a structure theorem that is similar to (and relies on) the celebrated theorem of Jorgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic 3-manifold is much bigger than its volume. Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves.

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arXiv2010

Invariant Heegaard Surfaces in Manifolds with Involutions and the Heegaard Genus of Double Covers

Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover π:M \to N we obtain the following bound on the Heegaard genus of N: g(N) \leq 4g(S) - 3. We also get a bound on the complexity of the branch set in terms of g(S). If we assume that M is non-Haken, by Casson and Gordon we may replace g(S) by g(M) in all the statements above.

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arXiv2009

Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture

In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open.

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arXiv2009

A proof of Waldhausen's uniqueness of splittings of S^3 (after Rubinstein and Scharlemann)

In [Topology 35 (1996) 1005--1023] J H Rubinstein and M Scharlemann, using Cerf Theory, developed tools for comparing Heegaard splittings of irreducible, non-Haken manifolds. As a corollary of their work they obtained a new proof of Waldhausen's uniqueness of Heegaard splittings of S^3. In this note we use Cerf Theory and develop the tools needed for comparing Heegaard splittings of S^3. This allows us to use Rubinstein and Scharlemann's philosophy and obtain a simpler proof of Waldhausen's Theorem. The combinatorics we use are very similar to the game Hex and requires that Hex has a winner. The paper includes a proof of that fact (Proposition 3.6).

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