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Xingru Zhang

Publications and source records attributed to Xingru Zhang.

At least 19 recordsLinked to original sources
arXiv2026

The AJ conjecture and connected sums of torus knots

The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and $a$ have the same sign. Notably, in cases where $pq=ab$ but $p\ne a$, the recurrence polynomial $\alpha(t,M,L)$ of $T(p,q)\#T(a,b)$ has repeated factors involving the variable $L$ after evaluation at $t=-1$. These appear to be the first examples of knots exhibiting this phenomenon. Therefore, the AJ conjecture requires a slight modification to accommodate this possibility.

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arXiv2021

Dehn fillings of knot manifolds containing essential twice-punctured tori

We show that if a hyperbolic knot manifold $M$ contains an essential twice-punctured torus $F$ with boundary slope $β$ and admits a filling with slope $α$ producing a Seifert fibred space, then the distance between the slopes $α$ and $β$ is less than or equal to $5$ unless $M$ is the exterior of the figure eight knot. The result is sharp; the bound of $5$ can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the $α$-filling contains no non-abelian free group. The proofs are divided into the four cases $F$ is a semi-fibre, $F$ is a fibre, $F$ is non-separating but not a fibre, and $F$ is separating but not a semi-fibre, and we obtain refined bounds in each case.

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arXiv2021

Characterizing slopes for torus knots, II

A slope $\frac pq$ is called a characterizing slope for a given knot $K_0\subset S^3$ if whenever the $\frac pq$--surgery on a knot $K\subset S^3$ is homeomorphic to the $\frac pq$--surgery on $K_0$ via an orientation preserving homeomorphism, then $K=K_0$. In a previous paper, we showed that, outside a certain finite set of slopes, only the negative integers could possibly be non-characterizing slopes for the torus knot $T_{5,2}$. Applying recent work of Baldwin--Hu--Sivek, we improve our result by showing that a nontrivial slope $\frac pq$ is a characterizing slope for $T_{5,2}$ if $\frac pq>-1$ and $\frac pq\notin \{0,1, \pm\frac12,\pm\frac13\}$. In particular, every nontrivial L-space slope of $T_{5,2}$ is characterizing for $T_{5,2}$. As a consequence, if a nontrivial $\frac pq$-surgery on a non-torus knot in $S^3$ yields a manifold of finite fundamental group, then $|p|>9$.

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arXiv2019

Remarks on SU(2)-simple knots and SU(2)-cyclic 3-manifolds

We give some remarks on two closely related issues as stated in the title. In particular we show that a Montesinos knot is SU(2)-simple if and only if it is a 2-bridge knot, extending a result of Zentner for 3-tangle summand pretzel knots. We conjecture with some evidence that an SU(2)-cyclic rational homology 3-sphere is an L-space.

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arXiv2016

Finite Dehn surgeries on knots in $S^3$

We show that on a hyperbolic knot $K$ in $S^3$, the distance between any two finite surgery slopes is at most two and consequently there are at most three nontrivial finite surgeries. Moreover in case that $K$ admits three nontrivial finite surgeries, $K$ must be the pretzel knot $P(-2,3,7)$. In case that $K$ admits two noncyclic finite surgeries or two finite surgeries at distance two, the two surgery slopes must be one of ten or seventeen specific pairs respectively. For $D$-type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that $4m$ and $4m+4$ are characterizing slopes for the torus knot $T(2m+1,2)$ for each $m\geq 1$.

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arXiv2015

Character varieties, A-polynomials, and the AJ Conjecture

We establish some facts about the behavior of the rational-geometric subvariety of the $SL_2(\c)$ or $PSL_2(\c)$ character variety of a hyperbolic knot manifold under the restriction map to the $SL_2(\c)$ or $PSL_2(\c)$ character variety of the boundary torus, and use the results to get some properties about the A-polynomials and to prove the AJ conjecture for certain class of knots in $S^3$ including in particular any $2$-bridge knot over which the double branched cover of $S^3$ is a lens space of prime order.

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arXiv2014

Detection of knots and a cabling formula for A-polynomials

We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynomial. We also give a one-parameter family of infinitely many hyperbolic knots in $S^3$ each of which is detected by its knot Floer homology and $A$-polynomial. In addition we give a cabling formula for the A-polynomials of cabled knots in $S^3$, which is of independent interest. In particular we give explicitly the A-polynomials of iterated torus knots.

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arXiv2014

Dehn surgery on knots in $S^3$ producing Nil Seifert fibred spaces

We prove that there are exactly $6$ Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in $S^3$, with $\{60, 144, 156, 288, 300\}$ as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly $4$ specific hyperbolic knots in $S^3$ which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in $S^3$.

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arXiv2014

The AJ-conjecture and cabled knots over torus knots

We show that most cabled knots over torus knots in $S^3$ satisfy the AJ-conjecture, namely each $(r,s)$-cabled knot over each $(p,q)$-torus knot satisfies the $AJ$-conjecture if $r$ is not a number between $0$ and $pqs$.

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arXiv2012

Characterizing slopes for torus knots

A slope $\frac pq$ is called a characterizing slope for a given knot $K_0$ in $S^3$ if whenever the $\frac pq$-surgery on a knot $K$ in $S^3$ is homeomorphic to the $\frac pq$-surgery on $K_0$ via an orientation preserving homeomorphism, then $K=K_0$. In this paper we try to find characterizing slopes for torus knots $T_{r,s}$. We show that any slope $\frac pq$ which is larger than the number $\frac{30(r^2-1)(s^2-1)}{67}$ is a characterizing slope for $T_{r,s}$. The proof uses Heegaard Floer homology and Agol--Lackenby's 6--Theorem. In the case of $T_{5,2}$, we obtain more specific information about its set of characterizing slopes by applying more Heegaard Floer homology techniques.

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arXiv2012

Characteristic submanifold theory and toroidal Dehn filling

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes $α, β$ on the boundary of a hyperbolic knot manifold $M$ has been verified in all cases other than small Seifert filling slopes. In this paper we verify it when $α$ is a small Seifert filling slope and $β$ is a toroidal filling slope in the generic case where $M$ admits no punctured-torus fibre or semi-fibre, and there is no incompressible torus in $M(β)$ which intersects $\partial M$ in one or two components. Under these hypotheses we show that $Δ(α, β) \leq 5$. Our proof is based on an analysis of the relationship between the topology of $M$, the combinatorics of the intersection graph of an immersed disk or torus in $M(α)$, and the two sequences of characteristic subsurfaces associated to an essential punctured torus properly embedded in $M$.

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arXiv2012

Dehn fillings of knot manifolds containing essential once-punctured tori

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $β$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $α$ produces a Seifert fibred manifold, then $Δ(α,β)\leq 5$. Furthermore we classify the triples $(M; α,β)$ when $\D(α,β)\geq 4$. More precisely, when $\D(α,β)=5$, then $M$ is the (unique) manifold $Wh(-3/2)$ obtained by Dehn filling one boundary component of the Whitehead link exterior with slope -3/2, and $(α, β)$ is the pair of slopes $(-5, 0)$. Further, $\D(α,β)=4$ if and only if $(M; α,β)$ is the triple $\displaystyle (Wh(\frac{-2n\pm1}{n}); -4, 0)$ for some integer $n$ with $|n|>1$. Combining this with known results, we classify all hyperbolic knot manifolds $M$ and pairs of slopes $(β, γ)$ on $\partial M$ where $β$ is the boundary slope of an essential once-punctured torus in $M$ and $γ$ is an exceptional filling slope of distance 4 or more from $β$. Refined results in the special case of hyperbolic genus one knot exteriors in $S^3$ are also given.

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arXiv2009

Quasi-Fuchsian Surfaces In Hyperbolic Link Complements

We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with closed immersed incompressible surfaces.

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arXiv2007

Reducible And Finite Dehn Fillings

We show that the distance between a finite filling slope and a reducible filling slope on the boundary of a hyperbolic knot manifold is at most one.

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arXiv2006

Heegaard splittings and virtually Haken Dehn filling II

We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, non-fibered knot manifolds in the cusped Snappea census. For each such manifold M, we compute a number c(M), such that, for any n>c(M), the n-fold cyclic cover of M is large.

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arXiv2006

Characteristic Subsurfaces, Character Varieties and Dehn Filling

We give new bounds for the distance between two exceptional filling slopes for a 1-cusped hyperbolic 3-manifold in several different situations. The distance between a reducible slope and a slope that produces a manifold with finite fundamental group is at most 2. The distance between a reducible slope and one that produces a very small manifold is also at most 2. The distance between a reducible slope and one which produces a manifold with a pi_1 injective torus is at most 4. The methods used involve both characteristic submanifold theory and the theory of the PSL(2,C) character variety.

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