Nielsen number is a knot invariant

Minimum Braids: A Complete Invariant of Knots and Links

Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic tables.

An Affine Index Polynomial Invariant of Virtual Knots

This paper describes a polynomial invariant of virtual knots that is defined in terms of an integer labeling of the virtual knot diagram. This labeling is seen to derive from an essentially unique structure of affine flat biquandle for flat virtual diagrams. The invariant is discussed in detail with many examples,including its relation to previous invariants of this type and we show how to construct Vassiliev invariants from the same data.

Quandle Homology Theory and Cocycle Knot Invariants

This paper is a survey of several papers in quandle homology theory and cocycle knot invariants that have been published recently. Here we describe cocycle knot invariants that are defined in a state-sum form, quandle homology, and methods of constructing non-trivial cohomology classes.

Cocycle Invariants and Oriented Singular Knots

We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called \emph{oriented singquandles} and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.

A free-group valued invariant of free knots

The aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups). (Some minor mistakes are corrected)

Knot invariants for rail knotoids

To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for rail knotoids based on these associations. We then translate them to invariants of rail isotopy for rail arcs.

Finite type invariants for knotoids

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complet…

Turaev genus, knot signature, and the knot homology concordance invariants

We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.

Some bounds for the knot Floer $τ$-invariant of satellite knots

We use four dimensional techniques to derive general bounds on the $τ$ invariant of a satellite knot in $S^{3}$.

Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants

We introduce two sequences of two-variable polynomials $\{ L^n_K (t, \ell)\}_{n=1}^{\infty}$ and $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$, expressed in terms of index value of a crossing and $n$-dwrithe value of a virtual knot $K$, where $t$ and $\ell$ are variables. Basing on the fact that $n$-dwrithe is a flat virtual knot invariant we prove that $L^n_K$ and $F^n_K$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L^n_K$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.

$L^2$-Alexander invariant for knots

This paper deals with the study of a new family of knot invariants: the $L^2$-Alexander invariant. A main result is to give a method of computation of the $L^2$-Alexander invariant of a knot complement using any presentation of default 1 of the knot group.

The Rasmussen invariant of a homogeneous knot

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader's theorem which states that a knot is positive if and only if it is homogeneous and strongly quasipositive.

定向纽结的两个完全不变量

Two complete invariants of oriented knots

本文构造了定向经典纽结的两个完全不变量，其中一个为无限集，另一个为有限集，并给出从纽结图有限步计算有限集及打结数的算法。

Graphical Constructions for the sl(3), so(3) and G2 Invariants for Virtual Knots, Virtual Braids and Free Knots

We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the…

On Invariants of Morse Knots

We define and study Vassiliev invariants for (long) Morse knots. It is shown that there are Vassiliev invariants which can distinguish some topologically equivalent Morse knots. In particular, there is an invariant of order 3 for Morse knots with one maximum that distinguishes two different representations of the figure eight knot. We also present the results of computer calculations for some invariants of low order. It turns out that for Morse knots with two maxima there is a Z/2-valued invariant of order 6 which is not a reduction of any integer-valued invariant.

Strongly invertible knots, invariant surfaces, and the Atiyah-Singer signature theorem

We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.

Invariants of Knot Diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.

Upsilon invariants of L-space cable knots

We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,Υ_K$ and $Υ_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.

A higher-order genus invariant and knot Floer homology

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $L^2$-signatures bound this invariant from below.

Hyperbolic L-space knots and their Upsilon invariants

For a knot in the 3-sphere, the Upsilon invariant is a piecewise linear function defined on the interval [0,2]. For an L-space knot, the Upsilon invariant is determined only by the Alexander polynomial of the knot. We exhibit infinitely many pairs of hyperbolic L-space knots such that two knots of each pair have distinct Alexander polynomials, so they are not concordant, but share the same Upsilon invariant. Conversely, we examine the restorability of the Alexander polynomial of an L-space knot from the Upsilon invariant through the Legendre-Fenchel transformation.

The Three Loop Isotopy and Framed Isotopy Invariants of Virtual Knots

This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop framed isotopy invariant, is a regular isotopy invariant of framed virtual knots. The properties of these invariants are investigated at length. In addition, we make precise the informal notion of ``analogue''. Using this formal definition, it is proved that a generalized three loop …

An involutive upsilon knot invariant

Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and provides bounds on the three-genus of a knot.

On computational aspects of two classical knot invariants

We look into computational aspects of two classical knot invariants. We look for ways of simplifying the computation of the coloring invariant and of the Alexander module. We support our ideas with explicit computations on pretzel knots.

Secondary Upsilon invariants of knots

The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented.

A transcendental function invariant of virtual knots

In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the zero polynomial.

A refinement of the first Vassiliev invariant can distinguish the orientation of knots

We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for all n-component string link satellites of a framed long knot to a candidate for distinguishing all classical knots.

Computations of Quandle Cocycle Invariants of Knotted Curves and Surfaces

State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.

Invariants of Legendrian knots

We present two different constructions of invariants for Legendrian knots in the standard contact space $\R^3$. These invariants are defined combinatorially, in terms of certain planar projections, and are useful in distinguishing Legendrian knots that have the same classical invariants but are not Legendrian isotopic.

Hyperfinite knots via the CJKLS invariant in the thermodynamic limit

We set forth a definition of hyperfinite knots. Loosely speaking, these are limits of certain sequences of knots with increasing crossing number. These limits exist in appropriate closures of quotient spaces of knots. We give examples of hyperfinite knots. These examples stem from an application of the Thermodynamic Limit to the CJKLS invariant of knots.

基于高斯图的平面类纽结三变量超越不变量

A three-variable transcendental invariant of planar knotoids via Gauss diagrams

该研究通过高斯图索引函数提出平面类纽结的三变量超越不变量，证明其为一阶Vassiliev不变量，并用于给出同伦平面类纽结Gordian距离的下界。

Fibred knots and twisted Alexander invariants

We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. We illustrate examples of knots that have trivial Alexander polynomials but do not have twisted Alexander invariants of fibred knots.

Higher-order finite type invariants of classical and virtual knots and unknotting operations

Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called $F$-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of $F$-order $\le n+1$ is strictly stronger than that of $F$-order $\le n$ and that of GPV-order $\le 2n+1$. To obtain…

Casson invariant of knots associated with divides

I present a formula for the Casson invariant of knots associated with divides. The formula is written in terms of Arnold's invariants of pieces of the divide. Various corollaries are discussed.

对角纽结与tau不变量

Diagonal knots and the tau invariant

本文证明所有可由对角网格图表示的纽结均为正纽结，并给出一个非环面纽结的实例，扩展了该表示类的已知范围。

On the additivity of the Thurston--Bennequin invariant of Legendrian knots

In this article, we consider the maximal value of the Thurston--Bennequin invariant of Legendrian knots which topologically represent a fixed knot type in the standard contact 3-space and we prove a formula of the value under the connected sum operation of knots.

Extending Van Cott's bounds for the $τ$-invariant of satellite knots

We generalize C. Van Cott's results on the $τ$ and $s$-invariants of cabled knots to apply to general satellite knots. This paper uses no Heegaard-Floer homology, relying on more geometric techniques.

Vassiliev Invariants of Virtual Legendrian Knots

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.

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…

Cobordism invariants for knots with two indices

We construct an invariant of virtual knots which is a sliceness obstruction and sensitive to the $Δ$-move. This invariants works if $\Z_{2}\oplus \Z_{2}$-index of chords is present.

A Sequence of Degree One Vassiliev Invariants for Virtual Knots

For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual …

Finite type invariants of knots via their Seifert matrices

We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial.

Finite order invariants for $(n,2)$-torus knots and the curve $Y^2=X^3+X^2$

We describe the algebra of finite order invariants on the set of all $(n,2)$-torus knots.

Logarithmic knot invariants arising from restricted quantum groups

We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.

Invariants of $2$-knots

This short survey, which was written to accompany a minicourse at the BIRS conference "Topology in dimension 4.5", concerns invariants of knotted $2$-spheres in $S^4$, also known as $2$-knots. It covers invariants extracted from the algebraic topology of the knot exterior, including Alexander invariants, the Farber-Levine pairing and Casson-Gordon invariants, as well as gauge theoretic and combinatorial invariants. Details are scarce and new results inexistant.

Commensurability of knots and L^2-invariants

We show that the L^2-torsion and the von Neumann rho-invariant give rise to commensurability invariants of knots.

Whitney towers and abelian invariants of knots

We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional algorithm for computing these invariants.

Knot invariants derived from quandles and racks

The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.

Upsilon不变量为凸函数的双曲纽结新族

New family of hyperbolic knots whose Upsilon invariants are convex

针对Borodzik和Hedden提出的Upsilon不变量为凸的纽结分类问题，本文采用$(1,1)$-纽结的组合方法计算纽结Floer复形，构造出无穷多互不协边且Upsilon不变量为凸的双曲纽结，丰富了该类纽结的实例。

Invariant character varieties of hyperbolic knots with symmetries

We study character varieties of symmetric knots and their reductions mod p. We observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry.

Invariants of Handlebody-Knots via Yokota's Invariants

We construct quantum $\mathcal{U}_q(\mathfrak{sl}_{\,2})$ type invariants for handlebody-knots in the 3-sphere $S^3$. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten-Reshetikhin-Turaev invariants.

Kashaev invariants of twice-iterated torus knots

We calculate the asymptotic behavior of the Kashaev invariant of a twice-itarated torus knot and obtain topological interpretation of the formula in terms of the Chern--Simons invariant and the twisted Reidemeister torsion.

D不变量对一类代数可切片纽结的可切片性的阻碍

D invariants obstruction to sliceness of a class of algebraically slice knots

本文利用二重分支覆盖的d不变量，证明了一组代数可切片纽结实际上是不可切片的，从而给出了可切片性的新阻碍。

Invariants of knot diagrams and relations among Reidemeister moves

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some $Ω_3$-moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other one must in general use $Ω_3$-moves of at least two different classes. To show this, knot diagram invariants that jump only under $Ω_3$-moves are introduced. Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, …

双环面的一个例子及通过其定义的长虚拟纽结的不变量

An example of biquandle and an invariant of long virtual knots defined through it

本文引入双环面的一个例子，定义长虚拟纽结的不变量，并给出两个可交换长虚拟纽结的必要条件。