Nafaa Chbili
We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period. The proof of the main result is based on computing the Yamada skein algebra of the solid torus then proving that this algebra injects into the Kauffman bracket skein algebra of the solid torus.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We show that the Kauffman bracket skein modules of certain manifolds obtained from integral surgery on a (2,2b) torus link are finitely generated, and list the generators for select examples.
In this paper we give an alternative basis, $\mathcal{B}_{\rm ST}$, for the Kauffman bracket skein module of the solid torus, ${\rm KBSM}\left({\rm ST}\right)$. The basis $\mathcal{B}_{\rm ST}$ is obtained with the use of the Tempereley--Lieb algebra of type B and it is appropriate for computing the Kauffman bracket skein module of the lens spaces $L(p, q)$ via braids.
We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern-Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, Gelca and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a de…
We show that the skein vector space of the 3-torus is finitely generated. We show that it is generated by 9 elements: the empty set, some simple closed curves representing the non null elements of the first homology group with coefficients in \Z_2, and a link consisting of two parallel copies of one of the previous non empty knots.
$\rm{PGL}_2$ skein algebras of small surfaces
本文研究小曲面（闭环面与一次穿孔环面）的$\rm{PGL}_2$ skein代数，给出显式有限表示、显式基及其乘法结构，并计算自然映射的核与代数中心，结果依赖于量子参数$q$。
Fast Algorithm for Multiplication on the Skein Algebra of One-hole Torus
针对单孔环面 skein 代数乘法复杂度高的问题，提出多项式时间算法并给出低交叉数情形的闭式公式，显著提升计算效率。
For a surface $F$, the Kauffman bracket skein module of $F \times [0,1]$, denoted $K(F)$, admits a natural multiplication which makes it an algebra. When specialized at a complex number $t$, nonzero and not a root of unity, we have $K_t(F)$, a vector space over $\mathbb{C}$. In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space $K_t(T^2)$ has five distinct traces. One trace, the Yang-Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on $K_t(T^2)$ correspond to each of the four $\mathbb{Z}_2$ ho…
We compute the action of the $\mathfrak{gl}_2$-skein algebra of the torus on the $\mathfrak{gl}_2$-skein module of the solid torus. As a result, we show that the $\mathfrak{gl}_2$-skein modules of lens spaces is spanned by $\left(\left\lfloor{\frac{p}{2}}\right \rfloor+1\right)\left(2\left\lfloor{\frac{p}{2}}\right \rfloor+1\right)$ elements.
Carrega has shown that the Kauffman bracket skein module of the 3-torus over the field of rational functions in the variable A can be generated by 9 skein elements. We show this set of generators is linearly independent.
We compute the Kauffman skein module of the complement of torus knots in S^3. Precisely, we show that these modules are isomorphic to the algebra of Sl(2,C)-characters tensored with the ring of Laurent polynomials.
We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein module of a cylinder over the torus.
This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system.
We prove that, in order to derive the HOMFLYPT skein module of the lens spaces $L(p,1)$ from the HOMFLYPT skein module of the solid torus, $\mathcal{S}({\rm ST})$, it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant $X$ for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set $Λ^{aug}$, augmenting the basis $Λ$ of $\mathcal{S}({\rm ST})$.
Kauffman bracket skein module of the connected sum of two solid tori
本文确定了两个 genus 一 handlebody 连通和的 Kauffman 括号 skein 模块结构，证明了猜想，为计算任意连通和的模块奠定基础。
We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that is a monomial subalgebra of a quantum torus. We utilize the former method to generalize the Chebyshev homomorphism of Bonahon and Wong between skein algebras of surfaces to a Chebyshev-Frobenius homomorphism between skein modules of marked 3-manifolds, in the course of which we …
The Gilmer-Masbaum map is not injective on the Kauffman bracket skein module
本文通过计算 2-环面映射环面族的求值映射像，发现 Gilmer-Masbaum 映射在某些同调类上非单射，回答了其维数下界问题，并给出了 Dehn 扭转幂映射环面的 Kauffman 括号 skein 模的基。
We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combining these two types of DAHA, we construct the DAHA representation for the skein algebra on a genus-two surface, and we propose a DAHA polynomial for a double-torus knot, which is a simple closed curve on a genus two Heegaard surface in S3. Discussed is a relationship between the D…
Skein module dimensions of mapping tori of the 2-torus
本文研究了 2- торус 映射射影的 skein 模维数，通过分解扭曲 Hochschild 同调得到通用 q 下的维数计算。
We show that the if a sequence of normalized polynomials gives rise to a positive basis of the skein algebra of a surface, then it is sandwiched between the two types of Chebyshev polynomials. For the closed torus, we show that the normalized sequence of Chebyshev polynomials of type one $(\hat{T}_n)$ is the only one which gives a positive basis.
The skein partition function of the mapping torus
本文研究了映射圆柱的 skein 分割函数，通过生成函数和欧拉乘积展开计算了 $\text{GL}_N$-skein 模块的维度。
Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces $L(p,q)$. For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for $p\geq 13$.
In this paper we give a new basis, $Λ$, for the Homflypt skein module of the solid torus, $\mathcal{S}({\rm ST})$, which was predicted by Jozef Przytycki, using topological interpretation. The basis $Λ$ is different from the basis $Λ^{\prime}$, discovered independently by Hoste--Kidwell \cite{HK} and Turaev \cite{Tu} with the use of diagrammatic methods. For finding the basis $Λ$ we use the generalized Hecke algebra of type B, $\textrm{H}_{1,n}$, defined by the second author in \cite{La2}, which is generated by looping elements and braiding elements and which is isomorphic to the affine Hecke …
We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces), describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representati…
Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
本文借助复三次曲面的高亏格Gromov–Witten镜像对称构造刻画4穿孔球面skein代数，并证明其与1穿孔环面bracelets基结构常数的强正性。
A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces
本文研究了环面的Roger-Yang skein代数与Kauffman括号 skein代数之间的关系，通过同态映射刻画了代数的不可约有限维表示，并计算了结构常数以支持正性猜想。
We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY skein relation - the HOMFLY deformed (p,q)-numbers with one fermionic parameter.
A note on double affine Hecke algebra for skein algebra on twice-punctured torus
本文通过 Iwahori--Hecke 算子的 Heegaard 对偶，为双穿孔环面上的 skein 代数构造了 $C^\vee C_1$ 型双仿射 Hecke 代数的推广，并证明其自同构对应 Dehn 扭转，同时给出经典极限的簇代数构造。
We prove a uniqueness result for finite-dimensional representations of the Kauffman skein algebra $\mathcal{S}_A(S)$ of a surface $S$, when $A$ is a root of unity and when the surface $S$ is a sphere with at most four punctures or a torus with at most one puncture. We show that, if two irreducible representations of $\mathcal{S}_A(S)$ have the same classical shadow and the same puncture invariants, and if this classical shadow is sufficiently generic in the character variety $\mathcal{X}_{SL_2(\mathbb{C})} (S)$, then the two representations are isomorphic.
Explicit representations and Azumaya loci of skein algebras of small surfaces
本文通过显式公式构造单孔环面与四孔球面 skein 代数在所有单位根处的有限维表示，并利用其可约性确定 Azumaya 轨迹，揭示了其与经典阴影簇光滑轨迹的包含关系。
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.
In this paper we compute the Kauffman bracket skein module of the complement of $(2, 2p+1)$-torus knots, $KBSM(T_{(2, 2p+1)}^c)$, via braids. We start by considering geometric mixed braids in $S^3$, the closure of which are mixed links in $S^3$ that represent links in the complement of $(2, 2p+1)$-torus knots, $T_{(2, 2p+1)}^c$. Using the technique of parting and combing, we obtain algebraic mixed braids, that is, mixed braids that belong to the mixed braid group $B_{2, n}$ and that are followed by their ``coset'' part, that represents $T_{(2, 2p+1)}^c$. In that way we show that links in $T_{(…
Skeins on tori
本文研究 $GL_N$、$SL_N$ 在一般量子参数下环面的 skein 不变量，给出 $T^3$ 模维数并刻画 $T^2$ 相对 skein 代数，尤其建立其与 DAHA 的同构或满射关系。
Stated Skein Theory and Double Affine Hecke Algebra Representations
本文通过标示skein理论扩展Kauffman括号skein代数，将其嵌入量子6-环面构造洛朗多项式模，从而发展A_1型双重仿射Hecke代数的表示理论，并从纽结补空间构建拓扑定义的模。
This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and odd closures, and define even and odd versions of the Kauffman bracket ideal. These even and odd Kauffman bracket ideals are used to obstruct even and odd tangle closures. Using a basis of Habiro's for the even Kauffman bracket skein module of the solid torus, we define bases for …
Basis for KBSM of fibered torus with multiplicity two exceptional fiber
本文构造了环面与圆周积的 Kauffman 括号 skein 模（KBSM）的一族基，并以此找到 $(β,2)$-纤维化环面 KBSM 的新基，为计算小 Seifert 纤维化 $3$-流形的 KBSM 奠定基础。
The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver descriptions of the geometry. Embedding into M-theory, a large class of dualities of 3d $\mathcal{N}=2$ theories associated to quivers is obtained. The multi-cover skein relation admits a compact formulation in terms of quantum torus algebras associated to the quiver and in this lang…
Let $Δ$ be a trivial knot in the three-sphere. For every finite cyclic group $G$ of odd order, we construct a $G$-equivariant Khovanov homology with coefficients in the filed $\F_{2}$. This homology is an invariant of links up to isotopy in $(S^{3},Δ)$. Another interpretation is given using the categorification of the Kauffman bracket skein module of the solid torus. Our techniques apply in the case of graphs as well to define an equivariant version of the graph homology which categorifies the chromatic polynomial. Keywords: Khovanov homology, group action, equivariant Jones polynomial, skein …
In this paper we work toward the Homflypt skein module of the lens spaces $L(p,1)$, $\mathcal{S}(L(p,1))$, using braids. In particular, we establish the connection between $\mathcal{S}({\rm ST})$, the Homflypt skein module of the solid torus ST, and $\mathcal{S}(L(p,1))$ and arrive at an infinite system, whose solution corresponds to the computation of $\mathcal{S}(L(p,1))$. We start from the Lambropoulou invariant $X$ for knots and links in ST, the universal analogue of the Homflypt polynomial in ST, and a new basis, $Λ$, of $\mathcal{S}({\rm ST})$ presented in \cite{DL1}. We show that $\math…
$\rm{PGL}_2$ webs at roots of unity
本文研究 $\rm{PGL}_2$ skein 理论在单位根处的蛛网，构造切比雪夫多项式以得到中心元素，并证明偶数阶单位根时环面 Frobenius 同态非单射。
Based on the presentation of the Kauffman bracket skein module of the torus given by the third author in previous work, Charles D. Frohman and Răzvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with re…
We formulate the holographic principle for knots and links. For the "space" of all knots and links, torus knots T(2m+1,2) and torus links L(2m,2) play the role of the "boundary" of this space. Using the holographic principle, we find the skein relation of knots and links with the help of the recurrence relation for polynomial invariants of torus knots T(2m+1,2) and torus links L(2m,2). As an example of the application of this principle, we derive the Jones skein relation and its generalization with the help of some variants of (q,p)-numbers, related with (q,p)-deformed bosonic oscillators.
We introduce the concept of tied links in the solid torus, which generalize naturally the concept of tied links in $S^3$ previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and subsequently we recover this invariant by using Jones' method over the bt-algebra of type $\mathtt{B}$ and the Markov trace defined on this.
For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of gl(2) foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the gl(2) skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunc…
A Reidemeister Theorem for Solid Ribbon Torus Links
针对焊接链环的Reidemeister刻画这一长期未解问题，本文提出了实心带状环面链环（R4中仅含带状奇点的浸入实心环面）的Reidemeister定理，在广义带状同痕下给出完整刻画。
Let $K= K(w,b,t)$ be a 1-bridge braid in a solid torus $V$, and let $γ$ be a $(p,q)$ curve on the torus $T = \partial V$ of the exterior $M_K$ of $K$. It will be shown that Dehn filling on $T$ along $γ$ produces a solid torus if and only if $p$ and $q$ satisfy one of four conditions determined by the parameters $(w,b,t)$ of the knot $K$. This solves the classification problem raised by Menasco and Zhang for such Dehn fillings.
The computational complexity of the solid torus core recognition problem
本文研究实心环面核心识别问题的计算复杂性，给出其属于 NP 的替代证明，并证明其属于 co-NP，同时推出 Hopf 链环识别问题同时属于 NP 与 co-NP。
We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagram can be fully recovered from its decorated Gauss diagram, and we use it to establish a characterization of the decorated Gauss diagrams of closed braids.
Generalized Double Affine Hecke Algebra for Double Torus
通过引入Hecke算子的Heegaard对偶推广C1型双重仿射Hecke代数，揭示了其与双环面skein代数的关系，并给出了对应Dehn扭转的代数自同构。
Kauffman bracket intertwiners and the volume conjecture
本文针对考夫曼括号缠结算子与体积猜想，通过代数嵌入显式计算闭环面缠结算子并证明其迹的上极限为零，还对负欧拉特征曲面的周期微分同胚提出并证明了相应极限为零的猜想。
Skein algebras and quantized Coulomb branches
本文为带边界低亏格曲面构造量子化 K-理论 Coulomb 分支，揭示其与 Kauffman 括号 skein 代数的联系，并提出一般猜想。
It is a basic question in contact geometry to classify all non-isotopic tight contact structures on a given 3-manifold. If the manifold has a boundary, we need also specify the dividing set on the boundary. In this paper, we answer the classification question completely for the case of a solid torus by writing down a closed formula for the number of non-isotopic tight contact structures with any given dividing set on the boundary of the solid torus. Previously, only a few special cases were known due to work by Honda.
Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but mathematically complicated, topological objects are Knots. We present in this papers several examples, both old and new, of regularity of algebraic invariants of knots. Our main invariants are the Jones polynomial (1984) and its generalizations. In the first section, we discuss the concept of a symmetric knot, and give one important example -- a torus knot. In the se…
Classification of small links in the unmarked solid torus
该研究提出分析无标记实心环面中纽结与链环的框架，证明多变量Alexander多项式为环境同胚不变量，给出低环绕数链环的Dehn扭转次数界，最终完成交叉数≤6的非分裂链环分类。
To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a study of codimension~2 singularities of link diagrams. For closed braids with a fixed number of strands, trace graphs can be recognized up to equivalence excluding one type of moves in polynomial time with respect to the braid length.
We show that in any triangulation of a solid torus, there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid torus. This theorem imposes restrictions on the possible Riemannian metrics on a solid torus. It also has applications in knot theory.
The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann that the BMW algebras are isomorphic to algebras of tangles in (disc $\times$ interval), modulo Kauffman skein relations. This isomorphism allows one to see very clearly certain properties of the BMW algebras -- for example existence of Markov traces and conditional expectations an…
For Legendrian links in the 1-jet space of $S^1$ we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kauffman polynomial estimate for the Thurston-Bennequin number as well as to the existence of an ungraded augmentation of the Chekanov-Eliashberg DGA. Parallel results involving the HOMFLY-PT polynomial and 2-graded generalized normal rulings are established.
A Quantum trace map for 3-manifolds
本文为具有环面边界的3-流形定义了基于理想三角剖分的量子迹映射，将Bonahon-Wong和Lê的曲面量子迹映射推广到三维。
We describe the deformation space of a solid torus with boundary modelled on convex ideal hyperbolic polyhedra. This deformation space is given by natural Gauss--Bonnet type inequalities on the dihedral angles. The result extends to solid tori with an arbitrary conical singularity along the core. Our method is to decompose the torus into tetrahedra and maximize a volume functional.
We show that for any Legendrian link $L$ in the $1$-jet space of $S^1$ the $2$-graded ruling polynomial, $R^2_L(z)$, is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R^2_L(z)$ as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, we show that this specialization may be interpreted as the standard inner product on the algebra of symmetric functions that is often identified with a certain subalgebra of the HOMFLY-PT skein module of the solid torus. In contrast to the $2$-graded case, we are able to use $0$-…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。