Cyclic and finite surgeries on Montesinos knots

Boundary slopes of some non-Montesinos knots

It is shown that there exist alternating non-Montesinos knots whose essential spanning surfaces with maximal and minimal boundary slopes are not realised by the checkerboard surfaces coming from a reduced alternating planar diagram.

Immersed surfaces and Seifert fibered surgery on Montesinos knots

We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if $\frac 1{q_1-1} + \frac 1{q_2-1} + \frac 1{q_3-1} \leq 1$ then a Montesinos knot $K(\frac{p_1}{q_1}, \frac{p_2}{q_2}, \frac{p_3}{q_3})$ admits no atoroidal Seifert fibered surgery.

The Slope Conjecture for 3-String Montesinos Knots

The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying certain conditions.

Boundary slopes and the numbers of positive/negative crossings for Montesinos knots

We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.

Montesinos纽结的Delta-解结数为一

Montesinos Knots with Delta-Unknotting Number One

本文研究$\Delta$-解结数为一的Montesinos纽结，提出其猜想性刻画，并给出不同$\Delta$-移动将纽结变为平凡纽结的实例。

Alternating Montesinos knots and Conjecture $\mathbb{Z}$

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that all the pretzel knots of the form $P(p,q,r)$ (not necessarily alternating) have property $\mathbb{Z}$.

A table of boundary slopes of Montesinos knots

This note corrects errors in Hatcher and Oertel's table of boundary slopes of Montesinos knots which have projections with 10 or fewer crossings.

Twisted Alexander polynomials of tunnel number one Montesinos knots

We calculate the twisted Alexander polynomials of all tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$-representations and obtain their leading coefficients and degrees. As a corollary, we get some interesting examples, that is, nonfibered knots with monic Alexander polynomials which have non-monic twisted Alexander polynomials.

Lower bounds on boundary slope diameters for Montesinos knots

In this paper, two lower bounds on the diameters of the boundary slope sets are given for Montesinos knots. One is described in terms of the minimal crossing numbers of the knots, and the other is related to the Euler characteristics of essential surfaces with the maximal/minimal boundary slopes.

Crossing number and diameter of boundary slope set of Montesinos knot

It is shown that the diameter of the boundary slope set is bounded from above by the twice of the minimal crossing number for a Montesinos knot.

Mutation invariance of the arc index for some Montesinos knots

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

The Slope Conjecture for a Family of Montesinos Knots

The Slope Conjecture relates the degree of the colored Jones polynomial to the boundary slopes of a knot. We verify the Slope Conjecture and the Strong Slope Conjecture for Montesinos knots $M(\frac{1}{r},\frac{1}{s-\frac{1}{u}},\frac{1}{t} )$ with $r,u,t$ odd, $s$ even and $u\leq-1$, $r<-1<1<s,t$.

Bounds on numerical boundary slopes for Montesinos knots

We give (1) an upper bound on the denominators of numerical boundary slopes and (2) an upper bound on the differences between two numerical boundary slopes, for Montesinos knot exteriors.

The Slope Conjecture for Montesinos knots

The Slope Conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of incompressible surfaces. Our aim is to prove the Slope Conjecture for Montesinos knots, and to match parameters of a state-formula for the colored Jones polynomial of such knots with the parameters that describe their corresponding incompressible surfaces via the Hatcher-Oertel algorithm.

Non-standard components of the character variety for a family of Montesinos knots

We show that there are Montesinos knots with $n+1\geq 4$ tangles whose character varieties contain arbitrarily many irreducible components of dimension $d$ for any $1\le d\le n-2$. Moreover, these irreducible components can be chosen so that the trace of the meridian is non-constant.

Obstructing Sliceness in a Family of Montesinos Knots

Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's diagonalization theorem and the fact that the 2-fold covers of Montisinos knots bound plumbing 4-manifolds, many of which are negative definite. Some of our examples include knots with signature 0 and square determinant.

The Number of Closed Essential Surfaces in Montesinos Knots with Four Rational Tangles

In the complement of a hyperbolic Montesinos knot with 4 rational tangles, we investigate the number of closed, connected, essential, orientable surfaces of a fixed genus $g$, up to isotopy. We show that there are exactly 12 genus 2 surfaces and $8ϕ(g - 1)$ surfaces of genus greater than 2, where $ϕ(g - 1)$ is the Euler totient function of $g - 1$. Observe that this count is independent of the number of crossings of the knot. Moreover, this class of knots form an infinite class of hyperbolic 3-manifolds and the result applies to all such knot complements.

Montesinos 纽结的 Delta-解纽数

Delta-Unknotting Number for Montesinos Knots

本文确定了特定 Montesinos 纽结族的 Delta-解纽数，证明其等于 Conway 多项式第二系数的绝对值，涵盖所有正 pretzel 纽结及部分双桥纽结。

The Montesinos trick for proper rational tangle replacement

Recently Iltgen, Lewark and Marino introduced the concept of a proper rational tangle replacement and the corresponding notion of the proper rational unknotting number. In this note we derive a version of the Montesinos trick for proper rational tangle replacement and use it to study knots with proper rational unknotting number one. We prove that knots with proper rational unknotting number one are prime and classify the alternating knots with proper rational unknotting number one. We also study Montesinos knots with proper rational unknotting number one.

Quasi-alternating Montesinos links

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as provided bounds on unknotting number and slice genus, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.

普查中非对称L-空间纽结的拟交错手术

Quasi-alternating surgeries on asymmetric L-space knots in the census

本文针对SnapPy普查中的9个非对称L-空间纽结，借助Montesinos技巧验证了此前通过计算机得到的每个纽结恰有两种拟交错手术的结论。

The ${\rm SL}(2,\mathbb{C})$-character variety of a Montesinos knot

For each Montesinos knot $K$, we propose an efficient method to explicitly determine the irreducible ${\rm SL}(2,\mathbb{C})$-character variety, and show that it can be decomposed as $\mathcal{X}_0(K)\sqcup\mathcal{X}_1(K)\sqcup\mathcal{X}_2(K)\sqcup\mathcal{X}'(K)$, where $\mathcal{X}_0(K)$ consists of trace-free characters, $\mathcal{X}_1(K)$ consists of characters of "unions" of representations of rational knots (or rational link, which appears at most once), $\mathcal{X}_2(K)$ is an algebraic curve, and $\mathcal{X}'(K)$ consists of finitely many points when $K$ satisfies a generic conditi…

Kauffman-Harary conjecture holds for Montesinos Knots

The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman-Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman-Harary conjecture for all Montesinos links. Finall…

All exceptional surgeries on alternating knots are integral surgeries

We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.

4股蒙特西诺斯结外皮的卡夫曼括号 skein 模中的 torsion

Torsion in Kauffman bracket skein module of a $4$-strand Montesinos knot exterior

该研究证明了4股蒙特西诺斯结外皮的卡夫曼括号 skein 模存在torsion，回答了Kirby列表中的一个疑问。

Cable knots do not admit cosmetic surgeries

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.

Ravels arising from Montesinos Tangles

A ravel is a spatial graph which is non-planar but contains no non-trivial knots or links. We characterize when a Montesinos tangle can become a ravel as the result of vertex closure with and without replacing some number of crossings by vertices.

On hyperbolic knots realizing the maximal distance between toroidal surgeries

For a hyperbolic knot in the 3-sphere, the distance between toroidal surgeries is at most 5, except the figure eight knot. In this paper, we determine all hyperbolic knots that admit two toroidal surgeries with distance 5.

Hyperbolic knots with three toroidal Dehn surgeries

It is conjectured that a hyperbolic knot admits at most three Dehn surgeries which yield closed three manifolds containing incompressible tori. We show that there exist infinitely many hyperbolic knots which attain the conjectural maximum number. Interestingly, those surgeries correspond to consecutive integers.

Knots in handlebodies with nontrivial handlebody surgeries

We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.

Non left-orderable surgeries on negatively twisted torus knots

We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non left-orderable fundamental groups.

Hyperbolic, L-space knots and exceptional Dehn surgeries

A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.

Connected sums of knots do not admit purely cosmetic surgeries

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.

Alternating odd pretzel knots and chirally cosmetic surgeries

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2,5) and (2,7)-torus knots, the genus 2 and 3 alternating odd pretzel knots do not admit any chirally cosmetic surgeries. Further, we show that for a fixed genus, at most finitely many alternating odd pretzel knots admit chirally cosmetic surgeries.

Purely cosmetic surgeries and pretzel knots

We show that all pretzel knots satisfy the (purely) cosmetic surgery conjecture, i.e. Dehn surgeries with different slopes along a pretzel knot provide different oriented three-manifolds.

Chirally Cosmetic Surgeries on Kinoshita-Terasaka and Conway knot families

In this note, we prove that a nontrivial Kinoshita-Terasaka or Conway knot does not admit chirally cosmetic surgeries, by calculating the finite type invariant of order 3.

The classification of exceptional Dehn surgeries on 2-bridge knots

We will classify all exceptional Dehn surgeries on 2-bridge knots according to whether they produce reducible, toroidal, or small Seifert fibered manifolds.

Networking Seifert Surgeries on Knots III

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a "hyperbolic seiferter" or a "hyperbolic annular pair" produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate …

非环面的正2桥结不接受镜像化妆品手术

A non-torus positive 2-bridge knot does not admit chirally cosmetic surgeries

该研究证明了非环面的正2桥结不接受镜像化妆品手术，通过Dehn手术得到反向定向同胚的3流形。

3-braid knots do not admit purely cosmetic surgeries

A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds. Using recent work of Hanselman, we show that (nontrivial) knots which arise as the closure of a 3-stranded braid do not admit any purely cosmetic surgeries.

Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any m which is not -2 it contains a successive triple of Seifert surgeries (K, m), (K, m +1), (K, m +2) on a hyperbolic k…

Satellite knots and L-space surgeries

We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots.

Toroidal Seifert fibered surgeries on alternating knots

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus knot and the surgery slope is 2(p+q) with |p|, |q| at least three.

Stair-step bridge spectra does not imply high distance

Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra but are not high distance, by computing the bridge spectra of certain pretzel knots and Montesinos knots.

Dehn surgeries on knots in product manifolds

We show that if a surgery on a knot in a product sutured manifold yields the same product sutured manifold, then this knot is a 0-- or 1--crossing knot. The proof uses techniques from sutured manifold theory.

同调球中纽结的美容手术与 Casson–Walker 不变量

Cosmetic surgeries on knots in homology spheres and the Casson--Walker invariant

针对同调球中纽结的美容手术问题，本文利用三维球内2分支链环的 Casson–Walker 不变量有理手术公式，给出纽结与手术斜率存在此类手术时的若干约束。

Seifert fibered surgeries on strongly invertible knots without primitive/Seifert positions

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.

Cosmetic surgeries on knots in $S^3$

Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in $S^3$, we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery formula for correction terms in Heegaard Floer homology.

Finite knot surgeries and Heegaard Floer homology

Extensive rewrite. Tables and proofs have been reformatted and/or rewritten for clarity.

Cosmetic surgeries on genus one knots

In this paper, we prove that there are no truly cosmetic surgeries on genus one classical knots. If the two surgery slopes have the same sign, we give the only possibilities of reflectively cosmetic surgeries. The result is an application of Heegaard Floer theory and number theory.

Representations of the (-2,3,7)-Pretzel Knot and Orderability of Dehn Surgeries

We construct a 1-parameter family of $\mathrm{SL}_2(\mathbf{R})$ representations of the pretzel knot $P(-2,3,7)$. As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6. Furthermore, we discuss a family of knots and exhibit similar orderability results for a few other examples.

A note on cabled slice knots and reducible surgeries

We consider the question of when a slice knot admits a reducible Dehn surgery. By analyzing the correction terms associated to such a surgery, we show that slice knots cannot admit surgeries with more than two summands. We also give a necessary Heegaard Floer theoretic condition for a positive cable of a knot to be slice.

Special alternating knots with sufficiently many twist regions have no chirally cosmetic surgeries

We show that a special alternating knot with sufficiently large number (more than $63$) of twist regions has no chirally cosmetic surgeries, a pair of Dehn surgeries producing orientation-reversingly homeomorphic $3$-manifolds. In the course of proof, we provide the optimal upper bounds of the primitive finite type invariants of degree 2 and 3 that solve Willerton's conjecture.

Pretzel knots with L-space surgeries

A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.