Melissa L. Macasieb, Thomas W. Mattman
Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n \neq 7, we show that M is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.
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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.
We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.
There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S, there is only a finite number of pretzel links whose Jones polynomials have span S. More concretely, we provide an algorithm useful for deciding whether or not a given knot is pretzel. As an application we identify all the pretzel knots up to nine crossings, proving in particular that $8_{12}$ is the first non-pretzel knot.
We prove that an odd pretzel knot is doubly slice if it has $2n+1$ twist parameters consisting of $n+1$ copies of $a$ and $n$ copies of $-a$ for some odd integer $a$. Combined with the work of Issa and McCoy, it follows that these are the only doubly slice odd pretzel knots.
If p/q > 18, p is odd, and $p/q \ne 37/2$, (p,q)-Dehn surgery for the (-2,3,7)-pretzel knot produces a 3-manifold without Reebless foliation.
We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for non-pretzel Montesinos knots of length three.
We show that for an alternating pretzel knot K the canonical genera of its Whitehead doubles W(K) are equal to the crossing number c(K) of K, verifying a conjecture of Tripp in the case of these knots.
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.
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.
We give a complete characterization of the topological slice status of odd 3-strand pretzel knots, proving that an odd 3-strand pretzel knot is topologically slice if and only if either it is ribbon or has trivial Alexander polynomial. (By work of [FS85], a nontrivial odd 3-strand pretzel knot $K$ cannot both be ribbon and have $Δ_K(t)=1$.) We also show that topologically slice even 3-strand pretzel knots (except perhaps for members of Lecuona's exceptional family of [Lec13]) must be ribbon. These results follow from computations of the Casson-Gordon 3-manifold signature invariants associated …
In this note we exhibit concrete examples of characterizing slopes for the knot $12n242$, aka the $(-2,3,7)$-pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the non-constructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.
In the present paper, we will show that a $(p,q,r)$-pretzel knot has the representativity 3 if and only if $(p,q,r)$ is either $\pm(-2,3,3)$ or $\pm(-2,3,5)$. We also show that a large algebraic knot has the representativity less than or equal to 3.
We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy representations determine the genus and fiberedness of the knots.
In this paper, we compute the Khovanov homology over \Q for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over \Q and \Z). Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.
The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable subsurfaces bounded by the knot. In this paper, we determine the crosscap numbers of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos knots developed by Hatcher and Oertel.
We prove that many pretzel knots of the form $P(2n,m,-2n\pm1,-m)$ are not topologically slice, even though their positive mutants $P(2n, -2n\pm1, m, -m)$ are ribbon. We use the sliceness obstruction of Kirk and Livingston related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.
A pretzel knot $K$ is called $odd$ if all its twist parameters are odd, and $mutant$ $ribbon$ if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5-stranded pretzel knots are $mutant$ $ribbon$. We do this in stages by first showing that 5-stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group, and thus in the smooth knot concordance group as well. N…
We compute the involutive knot invariants for pretzel knots of the form P(-2,m,n) for m and n odd and greater than or equal to 3.
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.
We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not binary dihedral.
We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it admits no non-trivial cyclic or finite surgeries. We also summarize similar results for other (-3,3,n) pretzel knots including the observation that the Seifert surgeries for these knots are precisely those integral slopes lying between two of the boundary slopes.
Using the Hatcher-Oertel algorithm for finding boundary slopes of Montesinos knots, we prove the Slope Conjecture and the Strong Slope Conjecture for a family of 3-tangle pretzel knots. More precisely, we prove that the maximal degrees of the colored Jones polynomial of such knots determine a boundary slope as predicted by the Slope Conjecture, and that the linear term in the degrees correspond to the Euler characteristic of an essential surface.
We prove the non-left-orderability of the fundamental group of the $n$-th fold cyclic branched cover of the pretzel knot $P(3,-3,-2k-1)$ for all integers $k$ and $n\ge 1$. These $3$-manifolds are $L$-spaces discovered by Issa and Turner.
Explicit Formulas for the Alexander Polynomial of Pretzel Knots
本文给出了 pretzel 结绳的 Alexander 多项式显式公式，并推导出若干结论，包括 trivial Alexander 多项式的 pretzel 结绳特征。应用此结果，构造了一类拓扑上可割但不光滑可割的结。
In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a $(-2,3,2s+1)$-pretzel knot ($s\ge 3$) with slope $\frac{p}{q}$ is not left orderable if $\frac{p}{q}\ge 2s+3$, and that it is left orderable if $\frac{p}{q}$ is in a neighborhood of zero depending on $s$.
We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Gordon.
We give a complete description of exceptional surgeries on pretzel knots of type $(-2, p, p)$ with $p \ge 5$. It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a twisted $I$-bundle over the Klein bottle. We show that the other is homeomorphic to the one obtained by certain Dehn filling on the magic manifold. On the other hand, we show that all such pretzel knots admit no Seifert fibered surgeries.
We apply the theory of "peculiar modules" for the Floer homology of 4-ended tangles developed by Zibrowius (specifically, the immersed curve interpretation of the tangle invariants) to compute the Knot Floer Homology ($\widehat{HFK}$) of 3-stranded pretzel knots of the form ${P(2a,-2b-1,\pm(2c+1))}$ for positive integers $a,b,c$. This corrects a previous computation by Eftekhary; in particular, for the case of ${P(2a,-2b-1,2c+1)}$ where $b<c$ and $b<a-1$, it turns out the rank of $\widehat{HFK}$ is larger than that predicted by that work.
Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots
本文利用 Mayland 技术证明若干亏格一及高亏格环面链环结的纽结群双可序，首次给出非纤维化或非交错纽结的双可序例子。
We classify Dehn surgeries on (p,q,r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Agol and Lackenby's 6-theorem reduces the argument to knots with small indices p,q,r. We treat these using the Culler-Shalen norm of the SL(2,C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety.
Let K_s be a (-2,3,2s+1)-type Pretzel knot (s >= 3) and E(K_s)(p/q) be a closed manifold obtained by Dehn surgery along K_s with a slope p/q. We prove that if q>0, p/q >= 4s+7 and p is odd, then E(K_s)(p/q) cannot contain an R-covered foliation. This result is an extended theorem of a part of works of Jinha Jun for (-2,3,7)-Pretzel knot.
We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(s) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a central role.
In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative set of invariants of algebraic concordance - the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the ratio…
Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family $P(a,-a-2,-\frac{(a+1)^2}{2})$ where $a$ is an odd integer greater than $1$. Lecuona and Miller showed that $P(a,-a-2,-\frac{(a+1)^2}{2})$ are not slice unless $a\equiv 1, 11, 37, 47, 59 \pmod{60}$. In this note, we show that four-fifths of the remaining knots in the family are not slice.
Exceptional pretzel knots are not algebraically slice
本文证明Lecuona例外族pretzel纽结的Alexander多项式不满足Fox-Milnor条件，从而完成三股pretzel纽结的可切性分类，并移除相关分类中的非例外假设。
The JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one
本文研究沿亏格一经典扭结作0-手术所得3-流形的JSJ分解，并借助最小扭曲五链环异常填充的分类结果进行分析。
We compute the unknotting number of two infinite families of pretzel knots, $P(3,1,\dots,1,b)$ (with $b$ positive and odd and an odd number of 1s) and $P(3,3,3c)$ (with $c$ positive and odd). To do this, we extend a technique of Owens using Donaldson's diagonalization theorem, and one of Traczyk using the Jones polynomial, building on work of Lickorish and Millett.
Character varieties of even classical pretzel knots
本文确定了偶数经典pretzel结的不可约${\rm SL}(2,\mathbb{C})$表示的特征簇，并阐明了计算其A-polynomial的步骤。
We show that the SL(2,C)-character variety of the (-2,3,n) pretzel knot consists of two (respectively three) algebraic curves when 3 does not divide n (respectively 3 divides n) and give an explicit calculation of the Culler-Shalen seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgeries yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.
The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology of all 3-strand pretzel links, over the rational numbers.
Character varieties of odd classical pretzel knots
本文确定了奇经典 pretzel 纽结的 SL(2,C) 字符簇，并提出了计算其 A-多项式的方法。
We computed the arc index for some of the pretzel knots $K=P(-p,q,r)$ with $p,q,r\ge2$, $r\geq q$ and at most one of $p,q,r$ is even. If $q=2$, then the arc index $α(K)$ equals the minimal crossing number $c(K)$. If $p\ge3$ and $q=3$, then $α(K)=c(K)-1$. If $p\ge5$ and $q=4$, then $α(K)=c(K)-2$.
Quasipositivity and braid index of pretzel knots
本文利用Morton-Franks-Williams不等式和Khovanov-Rozansky concordance homomorphisms，计算了一条链索上具有偶数个交叉点的三股pretzel knots的braid index，并确定了其quasipositivity。
The pretzel knot $P(4, -3, 5)$ is not squeezed
本文通过比较Rasmussen不变量与Iida、Taniguchi提出的$q_M$不变量，证明无穷族三股pretzel扭结非挤压扭结，特别证得$P(4, -3, 5)$非挤压扭结，解答了Lewark2024年提出的问题。
We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of knots for which the non-sliceness is detected by the Alexander polynomial while several modern obstructions to sliceness vanish. The second family has the property that the correction terms from Heegaard-Floer homology of the double branched covers of these knots do not obstruct the existence of a rational homology ball; however, the Casson-Gordon invariants…
We show that the A-polynomial $A_n$ of the 1-parameter family of pretzel knots $K_n=(-2,3,3+2n)$ satisfies a linear recursion relation of order 4 with explicit constant coefficients and initial conditions. Our proof combines results of Tamura-Yokota and the second author. As a corollary, we show that the $A$-polynomial of $K_n$ and the mirror of $K_{-n}$ are related by an explicit $\GL(2,\BZ)$ action. We leave open the question of whether or not this action lifts to the quantum level.
We compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.
Delta-Unknotting Number for Pretzel Knots
本文计算了正 pretzel 结的 Δ-解结数，并给出了奇型正 pretzel 结和环面结的最大值，以及特定负型 pretzel 结的 Δ-解结数。
We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.
This paper exhibits an infinite family of hyperbolic knot complements that have three knot complements in their respective commensurability classes.
We provide a partial classification of the 3-strand pretzel knots $K = P(p,q,r)$ with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with $r$ even. We discover that there are only four possible subfamilies which may satisfy $u(K) = 1$. These families are determined by the sum $p+q$ and their signature, and we resolve the problem in two of these cases. Ingredients in our proofs include Donaldson's diagonalisation theorem (as applied by Greene), Nakanishi's unknotting bounds from the Alexander modu…
We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.
We consider the classical pretzel knots $P(a_1, a_2, a_3)$, where $a_1, a_2, a_3$ are positive odd integers. By using continuous paths of elliptic $\mathrm{SL}_2(\mathbb R)$-representations, we show that (i) the 3-manifold obtained by $\frac{m}{l}$-surgery on $P(a_1, a_2, a_3)$ has left orderable fundamental group if $\frac{m}{l} < 1$, and (ii) the $n^{\mathrm{th}}$-cyclic branched cover of $P(a_1, a_2, a_3)$ has left orderable fundamental group if $n > 2π/ \arccos(1-2/(1+a_1 a_2 + a_2 a_3 + a_3 a_1))$.
$\mathrm{SL}_2(\mathbb R)$-representations and left-orderable surgeries of $(-2, 3, 2n+1)$-pretzel knots
本文通过构造连续的$\mathrm{SL}_2(\mathbb R)$-表示，研究了$(-2, 3, 2n+1)$-pretzel结的群表示，并证明了特定条件下所得3-流形的fundamental group是左可排序的。
We show that the L^2-torsion and the von Neumann rho-invariant give rise to commensurability invariants of knots.
We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most $3$ hyperbolic knot complements in a cyclic commensurability class. Moreover if two hyperbolic knots have cyclically commensurable complements, then they are fibered with the same genus and are chiral. A characterisation of cyclic commensurability classes of complements of periodic knots is also given. In the non-periodic case, we reduce the characterisat…
We collect statistics which consist of the coefficients in the expansion of the generating polynomials that count the Kauffman states associated with certain classes of pretzel knots having n tangles, of r half-twists respectively.
We present a combinatorial method for a calculation of knot Floer homology with Z-coefficient of (1,1)-knots, and then demonstrate it for non-alternating (1,1)-knots with ten crossings and the pretzel knots of type (-2,m,n). Our calculations determine the unknotting numbers and 4-genera of the pretzel knots of this type.
Ribbonlength bounds for pretzel links and knots with $\leq 9$ crossings
本文研究了不超过9个交叉的结与结带链的结带长度界限，证明了结带链的最小折叠结带长度存在统一上界，并提供了相关数据表。
We calculate the universal character ring of a class of two-generator, one-relator groups. As an application we give a less technical proof of a result in [LT] on the universal character ring of the (-2,3,2n+1)-pretzel knot. We also give an elementary proof of a result in [Ma] on the character variety of the (-2,3,2n+1)-pretzel knot.
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