Jozef H. Przytycki
We discuss several open problems in classical Knot Theory and we develop techniques that allow us to study them: Lagrangian tangles, skein modules and Burnside groups.
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In a previous paper, Vértesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by gluing together $T_1, \dotsc, T_m$, then the knot Floer homology $\hat{\mathrm{HFK}}(L)$ of $L$ can be recovered from $\widetilde{\mathrm{CT}} (T_1), \dotsc, \widetilde{\mathrm{CT}} (T_m)$. In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floe…
We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology. We prove the basic properties of our homology. In particular, the short exact sequence of homologies of skein related tangles and the Kunneth formula for the tensor product of tangles.
In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams of crossing change, which we call the "genus-one morphism." It is proved that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain an invariant for singular tangles that extending sl(2) tangle homology; examples include Lee homology, Bar-Nat…
In this paper, we use skein-theoretic techniques to classify all virtual knot polynomials and trivalent graph invariants with certain smallness conditions. The first half of the paper classifies all virtual knot polynomials giving non-trivial invariants strictly smaller than the one given by the Higman-Sims spin model. In particular, we exhibit a family of skein theories coming from $\text{Rep}(O(2))$ with an interesting braiding. In addition, all skein theories of oriented virtual tangles with some smallness conditions are classified. In the second half of the paper, we classify all non-trivi…
Skein exact triangles in knot Floer homology
本文为链环Floer同调构造新的skein精确三角族，由有理缠结三元组描述，并建立框架以推广至任意有理缠结三元组。
4-dimensional Skein modules, Handle attachments, and Tangles
本文提出了一种四维 skein 模块在处理附件中的通用公式，基于 skein 模块在 $S^3 \times I $ 中的定义，并扩展了现有研究成果。
We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tangles embedded in a solid torus. We investigate the relations between the invariants constructed by evaluation representations (and affinization of them) and usual skein modules, and give tools for interpretations of annular skein modules as sub-algebras of intertwiners for partic…
This paper is based on my talks (`Skein modules with a cubic skein relation: properties and speculations' and `Symplectic structure on colorings, Lagrangian tangles and its applications') given in Kyoto (RIMS), September 11 and September 18 respectively, 2001. The first three sections closely follow the talks: starting from elementary moves on links and ending on applications to unknotting number motivated by a skein module deformation of a 3-move. The theory of skein modules is outlined in the problem section of these proceedings. In the first section we make the point that despite its long h…
We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a byproduct we prove the Montesinos-Nakanishi 3-move conjecture for 3-algebraic links (including 3-bridge links).
We apply the techniques of totally twisted Khovanov homology to the constructions by M. Asaeda, J. Przytycki, and A. Sikora of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe an invariant chain complex built out of resolutions with only non-contractible circles. We use these to understand the δ-graded homology for links with alternating projection to the surface.
We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented 3-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki's q-analogue of the first homology group of a 3-manifold. In certain cases, for example for links in submanifolds of rational homology 3-spheres, we will be able to defi…
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's $G_2$ quantum invariants. Our method consists in the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.
Fundamentals of cubic skein modules
本文针对现有扭结论研究中高次扭结模关注不足的问题，系统性研究三次扭结模，分析其在三维球面及任意三维流形中的结构性质，为高次扭结模研究搭建基础框架，有望催生新的纽结多项式不变量。
Quantum fundamental group of knot and its $SL_2$ representation
本文利用底部缠结构造纽结补空间的 skein 模，先建立量子表示的万有空间再按 skein 关系取商，并研究量子环面在边界上的作用，从而导出着色 Jones 多项式的 $A_q$ 多项式递推关系。
Talk 1: Open problems in knot theory that everyone can try to solve. Knot theory is more than two hundred years old; the first scientists who considered knots as mathematical objects were A.Vandermonde (1771) and C.F.Gauss (1794). However, despite the impressive grow of the theory, there are simply formulated but fundamental questions, to which we do not know answers. I will discuss today several such open problems, describing in detail the 20 year old Montesinos-Nakanishi conjecture. Our problems lead to sophisticated mathematical structures (I will describe some of them in tomorrows talks), …
Ozsvath and Szabo recently constructed an algebraically defined invariant of tangles which takes the form of a DA bimodule. This invariant is expected to compute knot Floer homology. The authors have a similar construction for open braids and their plat closures which can be viewed as a filtered DA bimodule over the same algebras. For a closed diagram, this invariant computes the Khovanov homology of the knot or link. We show that forgetting the filtration, our DA bimodules are homotopy equivalent to a suitable version of the Ozsvath- Szabo bimodules. In addition to giving a relationship betwe…
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.
Conway Rational Tangles and the Thompson Group
本文通过 Conway 有理缠绕的视角，研究如何系统地寻找 Thompson 群元素以生成特定纽结或链环，并给出了构造简单缠绕乘积或拼接的方法，为 skein 理论化的构造问题提供基础。
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。
With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which recovers link Floer homology $\operatorname{\widehat{HFL}}$. Moreover, we classify peculiar modules in terms of immersed curves on the 4-punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson, we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinte…
Frobenius homomorphisms for stated ${\rm SL}_n$-skein modules
本文证明陈述 SL_n-绞股模上的弗罗贝尼乌斯同态将带框纽结映射为约化幂初等多项式的穿线，从而推广 n=2 情形并证实相关猜想。
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