Mark E. Kidwell, Theodore B. Stanford
We give an upper bound on the z-degree of the Kauffman polynomial of a link, using bridges of length greater than one which are separated in some tangle decomposition of a link diagram. We construct some examples by wiring together rational tangles.
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In the present paper, we build a bridge between Conway-Coxeter friezes and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomials attached to rational links by using Conway-Coxeter friezes. As an application one can give a complete invariant on Conway-Coxeter friezes of zigzag-type.
Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define the Kauffman bracket ideal of T to be the ideal I_T generated by the reduced Kauffman bracket polynomials of all closures of T. If this ideal is non-trivial, then T does not embed in the unknot. We give an algorithm for computing a finite list of generators for the Kauffman brack…
We define polynomial tangle invariants $\nabla_T^s$ via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for $\nabla_T^s$ of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants $\nabla_T^s$ can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of $\nabla_T^s$: a Heegaard Floer homology $\widehat{\operatorname{HFT}}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\wid…
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 …
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.
This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket skein module is a linear combination of $O(2^g)$ basis elements, with each coefficient a polynomial with at most $n$ nonzero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by $C\sqrt{n}$ for some $C.$ From this it follows that the …
In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.
A note on the recursive computation of the bracket polynomial for closed 4-tangles
该研究针对4-缠结阴影，通过自身n次连接与特定闭合操作形成纽结，利用考夫曼4-股图幺半群基的状态矩阵递归计算所得纽结的考夫曼括号多项式。
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…
For any positive integer r, we exhibit a knot Kr with (20 $\times$ 2 r--1 + 1) crossings whose Jones polynomial V (Kr) is equal to 1 mod-ulo 2 r. Our construction rests on a certain 20-crossing tangle T 20 which is undetectable by the Kauffman bracket polynomial pair mod 2.
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…
We compute the Kauffman bracket polynomial of the numerator and denominator closures of A + A + ... + A ( A is repeated n times), where A is a 2-tangle shadow that has at most 4 crossings.
We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalization (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.
We use Kauffman's bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the invariant, and use it to compute several examples.
The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler measure of the Kauffman bracket. Second we find a general form for the Kauffman bracket of a link found by surgering in a single rational tangle along n unlinked components, all in a particular annulus. We then give a condition under which the Mahler measure of the Kauffman bracket of such families diverges. We give examples of the condition in action.
We compute the Kauffman bracket polynomial of the three-lead Turk's head, the chain sinnet and the figure-eight chain shadow diagrams. Each of these knots can in fact be constructed by repeatedly concatenating the same 3-tangle, respectively, then taking the closure. The bracket is then evaluated by expressing the state diagrams of the concerned 3-tangle by means of the Kauffman monoid diagram's elements.
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rationa…
We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-integer invariant of 4-tangles. Calculations are facilitated by viewing the determinant as the Kauffman bracket at a fourth root of -1, which sets the loop factor to zero. For rational tangles, our invariant coincides with the value of the associated continued fraction.
The bracket polynomial of the Celtic link shadow $CK_4^{2n}$
本文利用基于图形恒等式的递推关系和 4-tangle 代数框架两种互补方法，推导了 Celtic 链环 $CK_4^{2n}$ 阴影的 Kauffman 括号多项式。
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 the present paper, we will show that for any integer n>0 there are infinitely many twisted torus knots with n-string essential tangle decompositions.
We construct an infinite collection of knots with the property that any knot in this family has $n$-string essential tangle decompositions for arbitrarily high $n$.
In this paper, we introduce a notion called n/k-free tangle and study the degeneration ratio of tunnel numbers of knots.
This paper will be an exposition of the Kauffman bracket polynomial model of the Jones polynomial, tangle methods for computing the Jones polynomial, and the use of these methods to produce non-trivial links that cannot be detected by the Jones polynomial.
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauffman bracket skein module of I-bundles over surfaces. This is accomplished by borrowing ideas from Bar-Natan's Khovanov homology theory for tangles and cobordisms and using embedded surfaces to generate the chain groups, instead of diagrams.
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between our skein algebra and the one defined by Muller, and use it to show that the quantum trace map can be extended to the Muller skein algebra.
Tangle decompositions of alternating link complements
本研究聚焦素交错链环能否分解为两个素交错tangle的问题，改进已有结果证明该分解要么在交错链环图中可见，要么对应伪Montesinos链环，为交错链环结构研究提供新结论。
Knot Logic and Arborescent Links
该研究提出交叉代数用于计数树状纽结等的分量数，该代数关联布尔逻辑推广与图论基础，可应用于有理纽结、括号多项式及Khovanov同调相关研究。
Combinatorial knot Floer homology and tangle decompositions
该研究利用网格图确定对角纽结的组合纽结 Floer 同调，其特定项可检测素因子数量与缠分解，还将对角纽结与各类纽结作比较，为纽结理论研究提供了新方法。
We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in $\mathbb{Z}[A^2, A^{-2}, h]$. The decomposition of the resulting polynomial into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]h$ yields the decomposition of the Kauffman-Jones polynomial of virtual singular links into two components, one in $\mathbb{Z}[A^2, A^{-2}]$ and the other in $\mathbb{Z}[A^2, A^{-2}]A^2$, where both components are invariants for virtual singular links.
We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link $L$, the fundamental quandle of $L$ may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the fundamental quandle of periodic links, composite knots and satellite knots.
We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie algebra G_2. It is therefore related to G_2 in the same way that the HOMFLY polynomial is related to A_n and the Kauffman polynomial is related to B_n, C_n, and D_n. We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions.
Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequacy polynomial which is the invariant of alternating link families. Adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links whose generating link has at most $n=12$ crossings.
The multi-variable affine index polynomial was defined by the author in previous work. The aim of this short note is to update the definition so it is generalizable to virtual tangles and to show it is compatible with tangle decomposition. We also introduce the Turaev moves for virtual tangles, and discuss how to recover the weight of each crossing as an intersection pairing of homology classes.
We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.
We give an $SL_3$ analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Le. To any punctured bordered surface, we associate an $SL_3$ stated skein algebra which contains the $SL_3$ skein algebra of closed webs. These algebras admit natural algebra morphisms associated to the splitting of surfaces along ideal arcs. We give an explicit basis for the $SL_3$ stated skein algebra and show that the splitting morphisms are injective and describe their images. By splitting a surface along the edges of an ideal triangulation, we see that the $SL_3$ stated s…
We give an irreducible decomposition of the so-called local representations (see arXiv:0707.2151) of the quantum Teichmüller space $\mathcal{T}_q(Σ)$ where $Σ$ is a punctured surface of genus $g>0$ and $q$ is a primitive $N$-th root of unity with $N$ odd. As an application, we construct a family of representations of the Kauffman bracket skein algebra of the closed surface $\overlineΣ$.
We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution of two topological vertices and the HOMFLY-PT invariant of the 4-component link $L_{8n8}$, which both depend on four arbitrary representations. The key point is that both of these are related to the Hopf polynomials in composite representations, which are in turn expressed through composite Schur functions. The latter are further expres…
Planar decomposition of the HOMFLY polynomial for bipartite knots and links
本文提出了一种适用于由反平行锁缠结构成的二部纽结图的全新平面分解方法，将Kauffman括号理论从sl(2)推广到任意sl(N)，从而为HOMFLY多项式的计算提供了与树状演算互补的新途径，并开辟了通向同调技术的新方向。
Thin links and Conway spheres
本文提出缠结细性的相对版本，并借助 HFT 与 \widetilde{Kh} 浸入曲线不变量，通过沿 Conway 球面的缠结分解刻画细链环。
The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant of oriented (1,1) tangles, which is derived from the representation of the braid generator associated with the one-parameter family of four dimensional representations with highest weights (0,0|a) of the quantum superalgebra U_q[gl(2|1)]. We use an abstract tensor state model to evaluate the invariant, as per the construction of the bracket polynomial state model used by Louis Kauffman to derive the Jones polynomial. This model facilitates both computation and theoretical exploration. Our family of representations has…
Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials
本文提出一种改进的Goeritz方法，用于计算双部链接的HOMFLY-PT多项式，通过纯代数运算实现高效计算。
We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.
For any virtual link $L = S \cup T$ that may be decomposed into a pair of oriented $n$-tangles $S$ and $T$, an oriented local move of type $T \mapsto T'$ is a replacement of $T$ with the $n$-tangle $T'$ in a way that preserves the orientation of $L$. After developing a general decomposition for the Jones polynomial of the virtual link $L = S \cup T$ in terms of various (modified) closures of $T$, we analyze the Jones polynomials of virtual links $L_1,L_2$ that differ via a local move of type $T \mapsto T'$. Succinct divisibility conditions on $V(L_1)-V(L_2)$ are derived for broad classes of lo…
We adapt Thistlethwaite's alternating tangle decomposition of a knot diagram to identify the potential extreme terms in its bracket polynomial, and give a simple combinatorial calculation for their coefficients, based on the intersection graph of certain chord diagrams.
Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
本文将二分纽结图HOMFLY多项式的平面分解推广到（反）对称着色情形，通过投影恢复平面性，超越树状演算，并给出计算实例，揭示二分演化与反平行通道中R-矩阵特征值平方的分解。
Octahedral coordinates from the Wirtinger presentation
本研究针对由图D的Wirtinger生成元表出的、纽结群到SL₂(ℂ)的表示ρ，给出了其诱导的八面体分解几何参数的直接代数公式，并提供了判别ρ是否为Neumann-Zagier–Yokota势函数临界点的新显式准则。
Four-page index and linear upper bounds for ribbonlength
本文提出四页索引作为结的表示不变量，并证明ribbonlength的线性上界为2c(K)，改进了现有结果并提供图示估计方法。
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