Thang T. Q. Lê, Dylan P. Thurston, Tao Yu
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.
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On a computation of the skein tree depth of knots and links
本文研究纽结与链环的 skein 树深度，证明了新的上界与下界，并给出了迄今未确定的深度表，以度量 skein 关系计算链环多项式的复杂度。
Integer inequalities between knot invariants, skein tree depth and delta-crossing numbers
本文研究了结不变量、skein树深度和delta交叉数之间的整数不等式，通过理论和计算结果证明了新的上界，并给出了新的下界，同时提供了未确定的结和链的skein树深度表。
Consequences of the compatibility of skein algebra and cluster algebra on surfaces
本文通过证明曲面cluster代数与广义skein代数的相容性，解决Roger-Yang装饰Teichmüller空间形变量子化猜想，并证明正亏格曲面cluster代数非有限生成且不同于其上cluster代数。
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.
We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete link invariant are obtained.
This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Markov traces on Hecke algebras to the framework of HOMFLYPT Skein modules. Some open problems on singular Hecke algebras are also presented.
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, $H[R]$, $K[Q]$ and $D[T]$, based on the invariants of knots, $R$, $Q$ and $T$, denoting the reg…
Skein and cluster algebras of punctured surfaces
本文通过几何方法证明了带孔曲面的标记 skein 代数与上聚类代数的等价性，并展示了其在代数几何中的应用。
$\rm{PGL}_2$ 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$q$。
Quantum holonomic link invariants derived from stated skein algebras
本文基于带状态skein代数表示理论为标架链环定义了不变量，推导了其规范不变性，揭示了与量子群及量子双曲几何构造的联系，并给出了R矩阵显式公式以计算不变量。
We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., Lê T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges and we generalise internal skein algebras in this context. Now, one needs t…
In this paper we present two new bases, $B^{\prime}_{H_2}$ and $\mathcal{B}_{H_2}$, for the Kauffman bracket skein module of the handlebody of genus 2 $H_2$, KBSM($H_2$). We start from the well-known Przytycki-basis of KBSM($H_2$), $B_{H_2}$, and using the technique of parting we present elements in $B_{H_2}$ in open braid form. We define an ordering relation on an augmented set $L$ consisting of monomials of all different "loopings" in $H_2$, that contains the sets $B_{H_2}$, $B^{\prime}_{H_2}$ and $\mathcal{B}_{H_2}$ as proper subsets. Using the Kauffman bracket skein relation we relate $B_{…
Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and statistical mechanics. Algebraic topology based on knots may be characterized as a study of properties of manifolds by considering links (submanifolds) in a manifold and their algebraic structure. The main …
This paper defines several algebras associated to an oriented surface $S$ with a finite set of marked points on the boundary. The first is the skein algebra $Sk_q(S)$, which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product is given by superposition of links. A basis of this algebra is given, as well as several algebraic results. When $S$ is triangulable, the quantum cluster algebra $A_q(S)$ and quantum upper cluster algebra U_q(S) can be defined. These are algebras coming from the triangulations o…
Kauffman bracket skein algebra of the 4-holed disk
本研究基于秩4自由群的SL(2,ℂ)特征簇的相关洞见，给出了四穿孔圆盘的Kauffman括号skein代数的单项式基并找到其表示，推进了相关代数结构的研究。
It is natural to try to place the new polynomial invariants of links in algebraic topology (e.g. to try to interpret them using homology or homotopy groups). However, one can think that these new polynomial invariants are byproducts of a new more delicate algebraic invariant of 3-manifolds which measures the obstruction to isotopy of links (which are homotopic). We propose such an algebraic invariant based on skein theory introduced by Conway (1969) and developed by Giller (1982) as well as Lickorish and Millett (1987). (This is the first paper I wrote about skein modules, almost 20 years ago.…
Askey-Wilson polynomial and genus-two skein module
本文利用亏格二 skein 代数的 q-差分算子，建立了约化 Askey-Wilson 多项式与亏格二环柄体 skein 模之间的对应关系。
On positivity of Roger--Yang skein algebras
本研究将Kauffman括号skein代数的正性猜想推广到Roger--Yang skein代数，用第一类Chebyshev多项式等显式多项式给出正基候选并构成特定意义下的下界，还探讨了量子参数为复单位根时多项式与该代数中心的关系。
Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmuller space. In this paper, we consider surfaces with punctures which is not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang's Po…
Quantum duality maps, skein algebras and their ensemble compatibility
本文将带孔闭曲面的量子对偶映射推广至一般标记曲面，基于约化stated skein代数与量子迹映射研究其对偶兼容性，并给出Laurent表达式性质与结构常数正性的skein理论证明。
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…
Center of stated $\mathrm{SL}(n)$-skein algebras
本文研究陈述的 $\mathrm{SL}(n)$-skein 代数的中心性质，利用量子迹映射和唯一性定理，证明其有限生成性及 PI-度，并推广到量子模代数和量子簇代数的表示论。
Let $k$ be a subring of the field of rational functions in $x, v, s$ which contains $x^{\pm 1}, v^{\pm 1}, s^{\pm 1}$. If $M$ is an oriented 3-manifold, let $S(M)$ denote the Homflypt skein module of $M$ over $k$. This is the free $k$-module generated by isotopy classes of framed oriented links in $M$ quotiented by the Homflypt skein relations: (1) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$; (2) $L$ with a positive twist $=(xv^{-1})L$; (3) $L\sqcup O=(\frac{v-v^{-1}}{s-s^{-1}})L$ where $O$ is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the…
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。
Skein (3+1)-TQFTs from Non-Semisimple Ribbon Categories
本文基于绳结理论，定义了非半单有限单模绳结张量范畴的(3+1)维拓扑量子场论，提出'色非退化'条件，并推广了Crane-Yetter-Kauffman和Lyubashenko等不变量。
Stated SL(n)-Skein Modules and Algebras
本文建立带标记三维流形的 stated SL(n)-skein 模理论，证明切割同态、加厚曲面 skein 代数与量子群及辫张量结构的几何对应，并给出 factorization homology 等价和自由模基的结果。
We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra ${\cal N}=\Z/2\ltimes \ext *\R^2$ on the other side. We find…
A HOMFLYPT-type invariant for pseudo links via a resolution in Hecke algebras
本文通过Hecke代数中的解析构造了伪链接的HOMFLYPT型不变量，解决了伪 Reidemeister 1 移动带来的障碍，基于解析同态和Ocneanu迹得到自然的伪 skein 关系。
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。
C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Giller's invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. After modifying a 2-knot into a Montesinos twin in a natural way, we show that Giller's invariant is related to the Seiberg-Witten invariant of the exterior of the twin, glued to the complement of a fiber in E(2).
Bracelets bases are theta bases
本文证明标记曲面 skein 代数中的手镯基与 theta 基（含量子情形）重合，并由此推出强正性与原子性猜想。
Morse theory of loop spaces and Hecke algebras
本文通过定义基于基点组的多环路 $A_\infty$-代数，将 Morse 理论与辫子 skein 代数及 Hecke 代数联系起来，并证明其与环绕高维 Heegaard Floer $A_\infty$-代数等价，同时计算了 $S^2$ 情形的导出 Hecke 代数。
We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q(\mathfrak{sl}_3)$ and the $A_2$ clasps are projectors. Our categorification also gives a natural definition of a $q$-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.
Band bases as common triangular bases in cluster algebras from surfaces
本文证明无孔标记曲面的辫代数中，Thurston 构造的带基与量子簇代数中的 Kazhdan-Lusztig 型公共三角基一致，证实了 Thurston 的猜想，并提供了公共三角基存在的新实例。
We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebras, namely the full curl and the sum of the Murphy operators. We discuss their calculation also in terms of the framing factor associated to the appropriate irreducible representation of the quantum group SU(N,q).
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