Coloring $n$-String Tangles

The number of strings on essential tangle decompositions of a knot can be unbounded

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 $n$.

Coloring invariants of spatial graphs

We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.

The minimum crossing number of essential tangles

In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted strings.

On tangle decompositions of twisted torus knots

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.

通过着色检测quandles的非可接受性

Detecting non-admissibility of quandles via colorings

本文通过(1,1)结的着色提供判断quandles是否可接受的标准，构造了多个非可接受的quandles例子，为结不变量研究提供新方法。

Persistently laminar tangles

A persistent lamination for a knot K is an essential lamination in the complement of the K, which remains essential after every non-trivial Dehn surgery along K. In particular, this implies that all of the Dehn surgery manifolds have universal cover R^3. This paper presents a method for building tangles T with the property that every knot K obtained by tangle sum with T has a persistent lamination. A natural generalization to 2n-string tangles is also given.

The Trieste look at Knot Theory

This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox 3-coloring. We show also a relation between 3-colorings and the Jones polynomial. In the second part we develop the general theory of Fox colorings and show how to associate a symplectic structure to a tangle boundary so that tangles becomes Lagrangians (a proof of this result has not been published before). Chapter VI of the book "KNOTS: From combinatorics of kn…

Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links

Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is uniquely determined by $(B,T)$ and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating …

Galois action on knots II: Proalgebraic string links and knots

We discuss an action of the Grothendieck-Teichmüller proalgebraic group on the linear span of proalgebraic tangles, oriented tangles completed by a filtration of Vassiliev. The action yields a motivic structure on tangles. We derive distinguished properties of the action particularly on proalgebraic string links and on proalgebraic knots which can not be observed in the action on proalgebraic braids. By exploiting the properties, we explicitly calculate the inverse image of the trivial (the chordless) chord diagram under the Kontsevich isomorphism.

Alexander-Conway invariants of tangles

We consider an algebra of (classical or virtual) tangles over an ordered circuit operad and introduce Conway-type invariants of tangles which respect this algebraic structure. The resulting invariants contain both the coefficients of the Conway polynomial and the Milnor's mu-invariants of string links as partial cases. The extension of the Conway polynomial to virtual tangles satisfies the usual Conway skein relation and its coefficients are GPV finite type invariants. As a by-product, we also obtain a simple representation of the braid group which gives the Conway polynomial as a certain twis…

关于中间quandle着色链环不变量与因果性检测

On Medial Quandle Coloring Link Invariant and Detecting Causality

本文研究中间quandle着色链环不变量在(2+1)维全局双曲时空中检测因果性的能力，证明在特定等价关系下无法区分关键链环，并给出充分条件及广义定理。

Computation of quandle 2-cocycle knot invariants without explicit 2-cocycles

We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding $2$-cocycles. This permits the construction of many $2$-cocycle invariants without exhibiting explicit $2$-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann's knot coloring polynomial. Computatio…

Random Walk Invariants of String Links from R-Matrices

We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed.

Longitudinal Mapping Knot Invariant for SU(2)

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then this invariant can be thought of a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian-longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for gene…

Knot polynomials and generalized mutation

The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the present paper is a generalization of Conway's mutation of knots and links. Instead of flipping a 2-strand tangle, one flips a many-string tangle to produce a generalized mutant. In the presence of rotational symmetry in that tangle, the result is called a "rotant". We show that if a rotant is sufficiently simple, then its Jones polynomial agrees with that of the …

拓扑观点下的交叉点与弧

The crossing and the arc from the topological viewpoint

本文从拓扑视角研究纽结图元素，将弧和交叉点的等价类描述为探针的同痕类，探讨其同伦类在曲面tangle着色代数结构中的基础作用，并引入multicrossing complex统一多种同调理论。

有理纽结与pretzel纽结的仿射表示

Affine representations of rational and pretzel knots

本文利用取值于奇异向量丛跨范畴的 TQFT，构造有理与 pretzel 缠绕及纽结群到 AGL(1,C) 的仿射表示，给出 Alexander 多项式闭式、零点界，并分析 t=-1 时的着色。

Mutant knots with symmetry

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m<6 in general, and for all choices of m when the satellite is based on a cable knot pattern. We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are dif…

Tunnel number degeneration under the connected sum of prime knots

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.

Kontsevich 不变量与 Grothendieck--Teichmüller 群对 2-组件字符串链的作用

The Kontsevich invariant and the action of the Grothendieck--Teichmüller group on $2$-component string links

本文研究了 Kontsevich 不变量对关联子选取的依赖性，并展示了 Grothendieck--Teichmüller 群对 2-组件字符串链代数作用的非平凡性。