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Ruifeng Qiu

Publications and source records attributed to Ruifeng Qiu.

15 recordsLinked to original sources
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.

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arXiv2024

A lower bound of the crossing number of composite knots

Let $c(K)$ denote the crossing number of a knot $K$, and let $K_1\#K_2$ denote the connected sum of two oriented knots $K_1$ and $K_2$. As a famous old question in knot theory, whether $c(K_1\#K_2)=c(K_1)+c(K_2)$ is still unsolved. In this paper, we show that for any nontrivial knot $K=K_1\#\cdots\#K_n$, where $K_1,\ldots,K_n$ are oriented knots, the following inequality holds \[c(K)>\frac{1}{16}(c(K_1)+\cdots+c(K_n)).\] The result improves a well-known lower bound of $c(K)$ given by Lackenby.

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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.

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arXiv2014

Angle structures and hyperbolic $3$-manifolds with totally geodesic boundary

This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic $3$-manifold with totally geodesic boundary has an ideal triangulation that admits angle structures.

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arXiv2013

On the degeneration of tunnel numbers under connected sum

We show that, for any integer $n\ge 3$, there is a prime knot $k$ such that (1) $k$ is not meridionally primitive, and (2) for every $m$-bridge knot $k'$ with $m\leq n$, the tunnel numbers satisfy $t(k\# k')\le t(k)$. This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel number under connected sum and meridionally primitive knots.

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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 $g$.

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arXiv2009

Degenerating slopes with respect to Heegaard distance

Let $M=H_{+}\cup_{S} H_{-}$ be a genus $g$ Heegaard splitting with Heegaard distance $n\geq κ+2$: (1) Let $c_{1}$, $c_{2}$ be two slopes in the same component of $\partial_{-}H_{-}$, such that the natural Heegaard splitting $M^{i}=H_{+}\cup_{S} (H_{-}\cup_{c_{i}} 2-handle)$ has distance less than $n$, then the distance of $c_{1}$ and $c_{2}$ in the curve complex of $\partial_{-}H_{-}$ is at most $3\mathfrak{M}+2$, where $κ$ and $\mathfrak{M}$ are constants due to Masur-Minsky. (2) Let $M^{*}$ be the manifold obtained by attaching a collection of handlebodies $\mathscr{H}$ to $\partial_{-} H_{-}$ along a map $f$ from $\partial \mathscr{H}$ to $\partial_{-} H_{-}$. If $f$ is a sufficiently large power of a generic pseudo-Anosov map, then the distance of the Heegaard splitting $M^{*}=H_{+}\cup (H_{-}\cup_{f} \mathscr{H})$ is still $n$. The proofs rely essentially on Masur-Minsky's theory of curve complex.

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arXiv2008

A proof of the Gordon Conjecture

A combinatorial proof of the Gordon Conjecture: The sum of two Heegaard splittings is stabilized if and only if one of the two summands is stabilized.

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arXiv2008

Additivity of Heegaard genera of bounded surface sums

Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded connected surface $F$ and $\partial_i$ be the component of $\partial M_i$ containing $F$. If $M_i$ has a high distance Heegaard splitting, then any minimal Heegaard splitting of $M$ is the amalgamation of those of $M^1, M^2$ and $M^*$, where $M^i=M_i\setminus\partial_i\times I$, and $M^{*}=\partial_1\times I\cup_{F} \partial_2\times I$. Furthermore, once both $\partial_i\setminus F$ are connected, then $g(M) = Min\bigl\{g(M_1)+g(M_2), α\bigr\}$, where $α= g(M_1) + g(M_2) + 1/2(2χ(F) + 2 - χ(\partial_1) - χ(\partial_2)) - Max\bigl\{g(\partial_1), g(\partial_2)\bigl\}$; in particular $g(M)=g(M_1)+g(M_2)$ if and only if $χ(F)\geq 1/2Max\bigl\{χ(\partial_1), χ(\partial_2)\bigr\}.$ The proofs rely on Scharlemann-Tomova's theorem.

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arXiv2007

Boundary reducible handle additions on simple 3-manifolds

Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus two. For a slope $γ$ on $F$, we denote by $M(γ)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $γ$ on $F$. In this paper, we shall prove that there is at most one separating slope $γ$ on $F$ so that $M(γ)$ is $\partial$-reducible.

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arXiv2006

Stabilizations of Reducible Heegaard Splittings

C. Gordon conjectured that a connected sum of two Heegaard splittings is stabilized if and only if one of the two factors is stabilized (Problem 3.91 in Kirby's problem list). In this paper, we shall prove this conjecture.

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arXiv2006

Handle additions producing essential surfaces

We construct a small, hyperbolic 3-manifold $M$ such that, for any integer $g\geq 2$, there are infinitely many separating slopes $r$ in $\partial M$ so that $M(r)$, the 3-manifold obtained by attaching a 2-handle to $M$ along $r$, is hyperbolic and contains an essential separating closed surface of genus $g$. The result contrasts sharply with those known finiteness results on Dehn filling, and it also contrasts sharply with the known finiteness result on handle addition for the cases $g=0,1$. Our 3-manifold $M$ is the complement of a hyperbolic, small knot in a handlebody of genus 3.

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arXiv2004

Heegaard Splittings of Boundary Reducible 3-Manifolds

In this paper, we shall prove that any Heegaard splitting of a $\partial$-reducible 3-manifold $M$, say $M=W\cup V$, can be obtained by doing connected sums, boundary connected sums and self-boundary connected sums from Heegaard splittings of $n$ manifolds $M_{1},..., M_{n}$ where $M_{i}$ is either a solid torus or a $\partial$-irreducible manifold. Furthermore, $W\cup V$ is stabilized if and only if one of the factors is stabilized.

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arXiv2004

Small knots and large handle additions

We construct a hyperbolic 3-manifold $M$ (with $\partial M$ totally geodesic) which contains no essential closed surfaces, but for any even integer $g> 0$ there are infinitely many separating slopes $r$ on $\partial M$ so that $M[r]$, the 3-manifold obtained by attaching 2-handle to $M$ along $r$, contains an essential separating closed surface of genus $g$ and is still hyperbolic. The result contrasts sharply with those known finiteness results for the cases $g=0,1$. Our 3-manifold $M$ is the complement of a simple small knot in a handlebody.

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