arXiv subjects

Yanqing Zou

Publications and source records attributed to Yanqing Zou.

10 recordsLinked to original sources

arXiv2026

Building Foliations from Heegaard Diagrams

Every closed orientable 3-manifold admits both a Heegaard splitting and a coorientable codimension-one foliation. We address the foliation realization problem: constructing a foliation directly from a Heegaard diagram of arbitrary genus. Using Gabai's sutured manifold theory, we introduce the notion of a meridional sutured handlebody and prove that every such handlebody decomposes, via basic sutured decomposition operations, into a Reeb component together with product disks. We then present a three-step construction: (i) building two meridional sutured handlebodies from a given Heegaard diagram, (ii) gluing them along compatible disk regions by reversing disk decompositions, and (iii) filling the remaining cavities with a meridional sutured handlebody and some Reeb components. The construction applies to Heegaard diagrams of any genus.

math.GT↗

arXiv2026

Genericity of hyperbolic 3-manifolds via Dehn surgery

A significant result by Lickorish and Wallace shows that every closed, orientable 3-manifold can be obtained from a Dehn surgery on one link in 3-sphere. As links and Dehn surgeries vary vastly in the universe, a question arises: how can we describe their properties in vague? We introduce a counting model on links and Dehn surgeries, and prove that under this model, (1) a randon link is hyperbolic; (2) a random 3-manifold is hyperbolic.

math.GT↗

arXiv2025

Nonembeddable contractible open manifolds arising from Whitehead doubling

We study a family of genus-one contractible open manifolds constructed by iterated Whitehead doubling from a nontrivial knot $K$ and an even half-twist $m$. For each such pair, we prove that the resulting contractible open $3$-manifold $W(K,m)$ does not embed as an open subset of any compact, locally connected and locally $1$-connected metric $3$-space. We also classify this family: $W(K,m)$ is homeomorphic to $W(K',m')$ if and only if $m=m'$ and $K$ is isotopic to $K'$. Thus the knot type and the half-twist parameter form a complete invariant for these nonembeddable contractible open manifolds. The proof combines the topology of ends with JSJ decompositions, hyperbolic pieces, and a rank estimate for iterated Whitehead doubled knot groups. The same method gives infinitely many pairwise non-homeomorphic higher-dimensional examples which embed in no compact, locally connected and locally $1$-connected metric space of the same dimension.

math.GT↗

arXiv2025

Comparing $h$-genera, Bridge-1 genera and Heegaard genera of knots

Let $h(K)$, $g_H(K)$, $g_1(K)$, $t(K)$ be the $h$-genus, Heegaard genus, bridge-1 genus, tunnel number of a knot $K$ in the $3$-sphere $S^3$, respectively. It is known that $g_H(K)-1=t(K)\leq g_1(K)\leq h(K)\leq g_H(K)$. A natural question arises: when do these invariants become equal? We provide the necessary and sufficient conditions for equality and use these to show that for each integer $n\geq 1$, the following three families of knots are infinite: \begin{eqnarray} A_{n}=\{K\mid t(K)=n<g_1(K)\}, B_{n}=\{K\mid g_1(K)=n<h(K)\}, C_{n}=\{K\mid h(K)=n<g_H(K)\}. \end{eqnarray} This result resolves a conjecture in \cite{Mo2}, confirming that each of these families is infinite.

math.GT↗

arXiv2023

Counting double cosets with application to generic 3-manifolds

We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. As an application, we confirm a conjecture of Maher that hyperbolic 3-manifolds are exponentially generic in the set of 3-manifolds built from Heegaard splitting using complexity in Teichmüller metric.

math.GR↗

arXiv2023

Finiteness of mapping class groups and Heegaard distance

We prove that the mapping class group of a Heegaard splitting with a distance of at least 3 is finite. However, we have constructed a counterexample with a distance of 2 that disproves this assertion. In addition, the fact that the mapping class group of a Heegaard splitting with a distance of at most 1 is infinite, when combined with our results, provides an answer to the question of the finiteness of mapping class groups as viewed from Heegaard distance.

math.GT↗

arXiv2020

On tunnel numbers of a cable knot and its companion

Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $ K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$. Though $t(K^{\star})\leq t(K)+1$, Example 1.1 in Section 1 shows that in some case, $ t(K^{\star})\leq t(K)$. So it is interesting to know when $t(K^{\star})= t(K)+1$. After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot $K^{\star}$ and its companion $K$, $t(K^{\star})\geq t(K)$; (2) if either $K$ admits a high distance Heegaard splitting or $p/q$ is far away from a fixed subset in the Farey graph, then $t(K^{\star})= t(K)+1$. Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

math.GT↗

arXiv2017

An upper bound on distance degenerate handle additions

We prove that for any distance at least 3 Heegaard splitting and a boundary component $F$, there is a diameter finite ball in the curve complex $\mathcal {C}(F)$ so that it contains all distance degenerate curves or slopes in $F$.

math.GT↗

arXiv2015

3-manifolding admitting locally large distance 2 Heegaard splittings

From the view of Heegaard splitting, it is known that if a closed orientable 3-manifold admits a distance at least three Heegaard splitting, then it is hyperbolic. However, for a closed orientable 3-manifold admitting only distance at most two Heegaard splittings, there are examples shows that it could be reducible, Seifert, toroidal or hyperbolic. According to Thurston's Geometrization conjecture, the most important piece of eight geometries is hyperbolic. Thus to read out a hyperbolic 3-manifold from a distance two Heegaard splittings is critical in studying Heegaard splittings. Inspired by the construction of hyperbolic 3-manifolds with a distance two Heegaard splitting [Qiu, Zou and Guo, Pacific J. Math. 275 (2015), no. 1, 231-255], we introduce the definition of a locally large geodesic in curve complex and furthermore the locally large distance two Heegaard splitting. Then we prove that if a 3-manifold admits a locally large distance two Heegaard splitting, then it is a hyperbolic manifold or an amalgamation of a hyperbolic manifold and a seifert manifold along an incompressible torus, i.e., almost hyperbolic, while the example in Section 3 shows that there is a non hyperbolic 3-manifold in this case. After examining those non hyperbolic cases, we give a sufficient and necessary condition for a hyperbolic 3-manifold when it admits a locally large distance two Heegaard splitting.

math.GT↗

arXiv2013

The Heegaard distances cover all non-negative integers

In this paper, we prove that (1) For any integers $n\geq 1$ and $g\geq 2$, there is a closed 3-manifold $M_{g}^{n}$ which admits a distance $n$ Heegaard splitting of genus $g$ except that the pair of $(g, n)$ is $(2, 1)$. Furthermore, $M_{g}^{n}$ can be chosen to be hyperbolic except that the pair of $(g, n)$ is $(3, 1)$. (2) For any integers $g\geq 2$ and $n\geq 4$, there are infinitely many non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings of genus $g$.

math.GT↗