The universal character ring of the (-2,2m+1,2n)-pretzel link

通过空间图的pretzel链环的推广

A Generalization of Pretzel Links via Spatial Graphs

本文通过空间图推广pretzel链环，构建无限多个不同ribbon knots，并利用jones多项式区分具有相同alexander多项式的链环。

交替扭结链环的枚举

The Enumeration of Alternating Pretzel Links

本文研究交替扭结链环的计数问题，给出按交叉数 $c$ 计算其总数 $\mathcal{P}(c)$ 的闭式公式，并通过数值计算揭示其指数增长规律。

The Thurston polytope for four-stranded pretzel links

In this paper we use Heegaard Floer link homology to determine the dual Thurston polytope for pretzel links of the form P(-2r_1-1, 2q_1, -2q_2, 2r_2+1) where r_i and q_i are positive integers. We apply this result to determine the Thurston norms of spanning surfaces for the individual link components, and we explicitly construct norm-realizing examples of such surfaces.

Pretzel knots up to nine crossings

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.

关于 $χ-$切片扭结链环

On $χ-$slice pretzel links

本文研究扭结链环的 $χ-$切片性，完全分类正、负情形并部分分类3股与4股情形，进而给出界定有理同调四维球的无穷多族Seifert纤维空间。

Pretzel links、mutation与slice-ribbon猜想

Pretzel links, mutation, and the slice-ribbon conjecture

本文利用3重分支覆盖和Donaldson对角化定理证明2分量pretzel link P(p,q,-p,-q)不是slice的，并由此证明了4股2分量pretzel links的slice-ribbon猜想。

Colored HOMFLY polynomials for the pretzel knots and links

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g+1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU_q(N) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many muta…

Pretzel链的辫子指标：综合研究，第二部分

The Braid Indices of Pretzel Links: A Comprehensive Study, Part II

本文作为pretzel链辫子指标综合研究的第二部分，针对类型3 pretzel链给出了辫子指标的确定方法，除少数例外可精确确定，例外情形也能限定在两个连续整数内，并猜想实际指标为其中较大的整数。

Pretzel链环的Quandle着色拟阵

Quandle coloring quivers of pretzel links

本文系统研究pretzel链环的二面体quandle着色及着色拟阵，确定3-pretzel情形，并在n为素数时给出4-pretzel及一般m-pretzel链环的着色数和拟阵结构，附完整证明。

The one-row colored $\mathfrak{sl}_{3}$ Jones polynomials for pretzel links

The colored $\mathfrak{sl}_{3}$ Jones polynomial $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ are given by a link and an $(n_{1}, n_{2})$-irreducible representation of $\mathfrak{sl}_{3}$. In general, it is hard to calculate $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ for an oriented link $L$. However, we calculate the one-row $\mathfrak{sl}_{3}$ colored Jones polynomials $J_{(n, 0)}^{\mathfrak{sl}_{3}}(P(α,β,γ);q)$ for three-parameter families of oriented pretzel links $P(α,β,γ)$ by using Kuperberg's linear skein theory by setting $n_{2}=0$. Furthermore, we show the existence of the tails …

关于扭结链环极端 Khovanov 同调的几何实现

On geometric realizations of the extreme Khovanov homology of pretzel links

本文通过构造扭结链环实极端 Khovanov 同调的显式几何实现，为 Przytycki–Silvero 关于任意链环图极端 Khovanov 同调无挠的猜想提供新的支持证据。

On the universal character ring and the character variety of the (-2,3,2n+1)-pretzel knot

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.

Pretzel 链环的辫指数：全面研究，第一部分

The Braid Indices of Pretzel Links: A Comprehensive Study, Part I

本文通过基于标准链环图 Seifert 圈分解的系统分类方法，研究并完全确定了所有 Type 1 和 Type 2 非交错 pretzel 链环的精确辫指数。

Arithmeticity and Hidden Symmetries of Fully Augmented Pretzel Link Complements

This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite families of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden symmetries grows linearly with volume. This second family realizes…

$(3,3,3,3)$-pretzel链环外部空间的Kauffman括号skein模

Kauffman bracket skein module of the $(3,3,3,3)$-pretzel link exterior

本文证明$(3,3,3,3)$-pretzel链环外部空间的Kauffman括号skein模非有限生成，否证了Detcherry 2021年的有限性猜想。

Pretzel 链环 $\oldsymbol{P(1,1,n)}$ 的 Alexander-Conway 与 Bracket 多项式

Alexander-Conway and Bracket Polynomials of Pretzel Links $\boldsymbol{P(1,1,n)}$

本文研究三股 pretzel 链环 P(1,1,n) 的 Alexander-Conway 与 Kauffman bracket 多项式，重新考察这两个经典多项式不变量，以深化对其拓扑和代数性质的理解。

结带长度界限与不超过9个交叉的结与结带链

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。

The Khovanov homology of 3-strand pretzels, revisited

We compute the reduced Khovanov homology of 3-stranded pretzel links. The coefficients are the integers with the "even" sign assignment. In particular, we show that the only homologically thin, non-quasi-alternating 3-stranded pretzels are P(-p,p,r) with p an odd integer and r greater than or equal to p (these were shown to be homologically thin by Starkston and Qazaqzeh).

Twisting quasi-alternating links

Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are quasi-alternating, and we determine the thickness of Khovanov homology for "most" pretzel links with arbitrarily many strands.

偶数经典Pretzel结的特征簇

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。

奇经典扭结的字符簇

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。

Finite surgeries on three-tangle pretzel knots

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.

Character varieties of some families of links

In this paper we consider some families of links, including (-2,2m+1,2n)-pretzel links and twisted Whitehead links. We calculate the character varieties of these families, and determine the number of irreducible components of these character varieties.

The universal character ring of some families of one-relator groups

We study the universal character ring of some families of one-relator groups. As an application, we calculate the universal character ring of two-generator one-relator groups whose relators are palindrome, and, in particular, of the (-2,2m+1,2n+1)-pretzel knot for all integers m and n. For the (-2,3,2n+1)-pretzel knot, we give a less technical proof of a result in [LT1] on its universal character ring, and an elementary proof of a result in [Ma] on the number of irreducible components of its character variety.

一类双曲3-辫的几何三角剖分

Geometric triangulations of a family of hyperbolic 3-braids

本文为$n\ge7$的$(-2,3,n)$-椒盐结与链环补构造拓扑三角剖分，利用Casson和Rivin定理直接证明其几何性，给出了该类双曲3-辫的直接证明。

Split link detection for sl(P) link homology in characteristic P

If $P$ is a prime number, we show that reduced $\mathfrak{sl}(P)$ link homology with coefficients in $\mathbf{Z}/P$ detects split links. The argument uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. When $P = 2$, we recover Lipshitz-Sarkar's split link detection result for Khovanov homology with $\mathbf{Z}/2$ coefficients.

Canonical components of character varieties of arithmetic two bridge link complements

The canonical components of SL_2-character varieties of arithmetic two bridge link groups are determined.

Borromean链环的${\rm SL}(2,\mathbb{C})$-特征簇

The ${\rm SL}(2,\mathbb{C})$-character variety of the Borromean link

本文确定了Borromean链环的不可约SL(2,C)特征簇，并给出了扭曲Alexander多项式在该簇上的函数公式。

Whitehead 链环的 ${\rm SL}(3,\mathbb{C})$-特征簇的对称切片

The symmetric slice of ${\rm SL}(3,\mathbb{C})$-character variety of the Whitehead link

本文研究 Whitehead 链环的 ${\rm SL}(3,\mathbb{C})$-特征簇，并刻画其对称切片上的一个 Zariski 开子集。

链环的约化外围系统的刻画

Characterization of the reduced peripheral system of links

本文研究了链环的约化外围系统，针对其在四个及以上分量时非完整不变量的局限，提出了两种等价链环的刻画方法：基于链环图的图示刻画和基于4维空间带状实体环面的拓扑刻画。

Character varieties of double twist links

We compute both natural and smooth models for the $SL_2(\mathbb C)$ character varieties of the two component double twist links, an infinite family of two-bridge links indexed as $J(k,l)$. For each $J(k,l)$, the component(s) of the character variety containing characters of irreducible representations are birational to a surface of the form $C\times \mathbb C$ where $C$ is a curve. The same is true of the canonical component. We compute the genus of this curve, and the degree of irrationality of the canonical component. We realize the natural model of the canonical component of the $SL_2(\math…

环面链环的特征簇

Character varieties of torus links

本文研究环面链环补空间基本群表示到代数群G的模空间几何，通过发展内蕴分层将其与底层环面纽结关联，并显式计算了G=SL_2(C)和SL_3(C)时的E-多项式，揭示了其与链环股数的意外关系。

怀特海德链及其推广的Dehn填充相关特征簇的交集研究

On Intersection of Character Varieties Associated to Dehn Fillings of the Whitehead Link and Its Generalization

本文以怀特海德链为示例，研究双曲2桥链补空间经不同Dehn填充后得到的广义曲线在典范特征簇曲面上的相交情况，利用扭曲Alexander多项式分类对应可约特征的交点，探讨其拓扑意义并推广至其他2桥链。

对数顶点算子代数的特征标与环面链环的着色不变量

Characters of logarithmic vertex operator algebras and coloured invariants of torus links

本文证明了 $(1,p)$ 单态与三重态顶点算子代数的 $\mathfrak{sl}_r$ 特征标可由特定环面链环的着色 $\mathfrak{sl}_r$ Jones 不变量取极限得到，建立了顶点代数与量子拓扑不变量之间的联系。

Finite Type Link Homotopy Invariants

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the pairwise linking numbers of the components.

Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two bridge link complements

Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two bridge link groups are determined. Special values of the zeta functions at s=0,1,2 are also investigated.

Conway spheres as ideal points of the character variety

We prove that any Bonahon-Siebenmann family of Conway spheres for a hyperbolic link is associated to an ideal point of the character variety of the link.

Khovanov homology and the Fukaya category of the traceless character variety for the twice-punctured torus

We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,δ)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,δ)$ is an isotopy invaria…

Identifying the Canonical Component for the Whitehead Link

In this paper we determine topologically the canonical component of the $SL_2(\mathbb{C})$ character variety of the Whitehead link complement.

$SL_2(\mathbb{C})$-Character Variety of a Hyperbolic Link and Regulator

In this paper, we study the $SL_2(\mathbb{C})$ character variety of a hyperbolic link in $S^3$. We analyze a special smooth projective variety $Y^h$ arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in $K_2({\mathbb C}(Y^h))$. By using the regulator map from $K_2$ to the corresponding Deligne cohomology, we get some variation formulae on some Zariski open subset of $Y^h$. From this we give some discussions on a possible parametrized volume conjecture for both hyperbolic links and kno…

Stratification of $\mathrm{SU}(r)$-character varieties of twisted Hopf links

We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $Γ_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometric description of the character variety, proving that this $\mathrm{SU}(2)$-character variety is a deformation retract of the larger $\mathrm{SL}(2,\mathbb{C})$-character variety, as conjectured by Flo…