Louis H. Kauffman, David E. Radford
This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework.
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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。
Virtualized Delta, sharp, and pass moves for oriented virtual knots and links
本文研究定向虚拟链环上的虚拟化 Delta、sharp 与 pass 移动，给出链环经这些局部移动等价的充要条件，并确立可达到的解结数下界。
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 generating set of Reidemeister moves of oriented virtual knots
针对有向虚拟纽结Reidemeister移动生成集研究不足的问题，研究枚举得到17种去冗余的有向虚拟Reidemeister移动，证明4元素子集为其生成集，简化了有向虚拟纽结不变量的验证工作。
Tabulation of knots up to five triple-crossings and moves between oriented diagrams
本文枚举了三交叉数不超过五的所有素纽结极小图表，推导了连接同一有向纽结三交叉图的有向移动极小生成集，并提出三交叉数下界与亚历山大多项式宽度相关的猜想。
The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links.
In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a more algebraic perspective, and develop a more detailed theory for them and their associated invariants.
In this paper we study embeddings of oriented connected closed surfaces in $\mathbb S^3$. We define a complete invariant, the fundamental span, for such embeddings, generalizing the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived and employed to study handlebody knots, bi-knotted surfaces, and chirality of knots. These invariants are capable to distinguish inequivalent handlebody knots and bi-knotted surfaces with homeomorphic complements. Particularly, we obtain an alternative proof of the inequivalence of Ishii et al.'s ha…
The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone knot diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.
The unoriented band unknotting numbers of torus knots
本文研究环面结的无向带解结数，利用无向纽结 Floer 同调的挠阶证明其与可由环面参数计算的 pinch 数相等。
Recognizing oriented knots from finite quotients with one marked element
该研究证明，由μ_Kλ_K共轭类标记的纽结群的profinite完备化可确定定向纽结的等价类，其方法结合了Xu的真性定理等，适用于卫星与复合纽结，且有限群上的Eisermann染色多项式可确定定向纽结。
On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots
该研究解决了吉川定向第五变换不可被其他变换替代的开放问题，扩展了浸入曲面链的代数工具包并开发了双quandle着色理论，还证明了无限多对群同构但结quandle不同的带边2-结。
In 1993 K. Habiro defined $C_k$-move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by $C_k$-moves if and only if they have the same Vassiliev invariants of order $\leq k-1$. In this paper we define Vassiliev invariant of type $(k_1,...,k_l)$, and show that, for $k=k_1+...+k_l$, two oriented knots are transformed into each other by $C_k$-moves if and only if they have the same Vassiliev invariants of type $(k_1,...,k_l)$. We introduce a concept ` band description of knots' and give a diagram-oriented proof of this theorem. When $k_1=...=k_l=…
In this paper, we enumerate the number of oriented rational knots and the number of oriented rational links with any given crossing number and minimum genus. This allows us to obtain a precise formula for the average minimal genus of oriented rational knots and links with any given crossing number.
We have two results. First, we give 96 generating sets oriented singular Reidemeister moves; it is an answer to a question by Bataineh, Khaled, Elhamdadi, and Hajij who give a generating set of oriented singular Reidemeister moves using their computation. Second, in the theory of plane curve and Legendrian knots introduced by V. I. Arnold, we select which moves survive as those of Legendrian singular knots and fronts diagrammatically and explicitly.
Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs
本文扩展Polyak关于纽结与链环图表定向Reidemeister移动极小生成集的研究，新增11个4元素极小生成集并证明共12个为全部可能，还确定空间三价图图表定向Reidemeister型移动的极小生成集含10个移动。
The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of the warping degree for all oriented alternating diagrams obtained from the knot projection. In this paper, we consider the maximal number of regions which share no crossings for a knot projection with a fixed crossing, and give lower bounds for the warping degree.
An Extension of the $sl(n)$ Polynomial to Knotted 4-Valent Graphs
本文提出了一种扩展$sl(n)$多项式用于结4价图的新不变量，并给出了相关Reidemeister型移动的最小生成集。
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.
We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.
In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group $\vec{F}$. In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an oriented knot/link $\vec{L}$, there exists an element $g$ in $\vec{F}$ whose closure $\vec{\mathcal{L}}(g)$ is $\vec L$.
We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.
We extend the quandle cocycle invariant to the context of stuck links. More precisely, we define an invariant of stuck links by assigning Boltzmann weights at both classical and stuck crossings. As an application, we define a single-variable and a two-variable polynomial invariant of stuck links. Furthermore, we define a single-variable and two-variable polynomial invariant of arc diagrams of RNA foldings. We provide explicit computations of the new invariants.
Electric group for knots and links
本文为定向纽结和链环提出电群的一般构造方法，并借助图着色刻画其到有限群的同态，进而引入与量子不变量思想相近的着色链环张量网络不变量。
Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots
针对Plotnick--Suciu构造及Suciu的替代构造，研究发现基本quandle等对象无法决定定向2-纽结外部空间的同伦类型，还明确了Tanaka--Taniguchi例子的二阶同伦模性质。
We introduce \textit{Kaestner brackets}, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle $2$-cocycle invariants and the classical biquandle brackets as special cases, coinciding with them for oriented classical knots and links but defining generally stronger invariants for oriented virtual knots and links. We provide examples to illustrate the computation of the new invariant and to show …
We introduce the warping matrix which is a new description of oriented knots from a viewpoint of warping degree.
We introduce several algebraic structures related to handlebody-knots, including $G$-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in $Y$-oriented spatial trivalent graph diagrams representing $S^1$-oriented handlebody-knots to obtain computable invariants for handlebody-knots and handlebody-links. In the case of $G$-families of biquandles, we enhance the counting invariant using the group $G$ to obtain a polynomial invariant of handlebody-knots.
Fox-Milnor condition for concordant knots in homology 3-spheres
本文证明定向同调3-球中切片型纽结的Alexander多项式满足Fox-Milnor条件，建立了该类空间中合痕纽结的Alexander多项式关系。
A refinement of the Q-Polynomial for twisted knots
本文提出了区分Q多项式无法区分的无限族扭结的$Q^z$多项式，推导了交叉变化公式等应用，证明其为一阶Vassiliev不变量。
We study RNA foldings and investigate their topology using a combination of knot theory and embedded rigid vertex graphs. Knot theory has been helpful in modeling biomolecules, but classical knots place emphasis on a biomolecule's entanglement while ignoring their intrachain interactions. We remedy this by using stuck knots and links, which provide a way to emphasize both their entanglement and intrachain interactions. We first give a generating set of the oriented stuck Reidemeister moves for oriented stuck links. We then introduce an algebraic structure to axiomatize the oriented stuck Reide…
Let $Σ_0,Σ_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset Σ_0 \times [0,1]$ and $K_1 \subset Σ_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a smoothly and properly embedded annulus $A$ in $W \times [0,1]$ such that $\partial W=Σ_1 \sqcup -Σ_0$ and $\partial A=K_1 \sqcup -K_0$. This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $K \subset Σ\times [0,1]$ admits a no…
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…
We generalize the notion of biquandles to psyquandles and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gröbner psyquandle invariants of oriented singular knots and links. We consider the relationship between Alexander psyquandle colorings of pseudolinks and p-colorings of pseudolinks. As a special case we define a generalization of the Alexander polynomial for oriented singular links and pseudolinks w…
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…
Homotopy index polynomials for knotoids
定义了定向曲面中纽结样的同伦指标多项式及同伦类型，研究其基本性质，表明平面纽结样的同伦多项式不比之前定义的指标多项式弱，且能检测球面纽结样的不可旋转性。
Knots and links in 2-complexes
该研究引入并探究二维复形中的纽结与链环，定义了对应的环绕数与Kauffman型括号多项式，并探讨了其与3流形中纽结链环的关联。
For an oriented knot $K$, we construct a functor from the category of pointed quandles to the category of quandles in three different ways. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariant of knots, and study their properties.
This paper concerns the Dehn surgery construction, especially those Dehn surgeries leaving the manifold unchanged. In particular, we describe an oriented 1-cusped hyperbolic 3-manifold X with a pair of slopes r_1, r_2 such that the Dehn filled manifolds X(r_1), X(r_2) are oppositely oriented copies of the lens space L(49,18), and there is no homeomorphism h of X such that h(r_1)=h(r_2).
Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant $t^{I\left( \mathcal{L} \right) }$ is constructed for a link $\mathcal{L}$, where $I$ is the abelian Chern-Simons action and $t$ a formal constant. For oriented knotted vortex lines, $t^{I}$ satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, $t^{I}$ satisfies the skein relations of the Kauffman bracket polynomial. As an example the bracket polynomials of trefoil knots are computed, and the Jones polynomial is constructed from the bra…
The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal
本文证明了几乎正结的Rasmussen不变量、4亏格和3亏格相等，并基于其几乎正结图确定了Rasmussen不变量，得出任何几乎正结均非齐次且不存在4亏格为1的几乎正结。
Isotopy and equivalence of knots in 3-manifolds
本文研究了3流形中结的同伦与等价性，证明了在素、闭合、定向的3流形中，等价结当且仅当保持定向的映射类群为平凡群。对于不可约的闭合定向3流形，作者证明了保持自由同伦类的定向保形映射与恒等映射同伦。在S¹×S²的情况下，给出了许多结的同伦类被Gluck扭变的例子。
We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number.
A virtual knot is an equivalence class of embeddings of $ S^1 $ into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of $ S^1 $ into a thickened surface $ Σ_g \times I $; what is the minimal genus of oriented surfaces $ S \hookrightarrow M \times I $ with the embedded $ S^1 $ as boundary, where $ M $ is an oriented $ …
Knotted vortices such as those produced in water by Kleckner and Irvine tend to transform by reconnection to collections of unknotted and unlinked circles. The reconnection number $R(K)$ of an oriented knot of link $K$ is the least number of reconnections (oriented re-smoothings) needed to unknot/unlink $K$. Putting this problem into the context of knot cobordism, we show, using Rasmussen's Invariant that the reconnection number of a positive knot is equal to twice the genus of its Seifert spanning surface. In particular an $(a,b)$ torus knot has $R = (a-1)(b-1).$ For an arbitrary unsplittable…
We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complement and knot Floer homology, and define intermediate Legendrian invariants.
This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding its structure in terms of compositions of morphisms, where elementary morphisms are associated to oriented classical and virtual crossings in the diagram.
A new algebraic method for computing helicity is developed, by discovering a relationship between helicity of fluid mechanics and algebraic polynomial invariants of knot theory. We have constructed a topological invariant $t^{H\left(\mathcal{L}\right)}$ for a link $\mathcal{L}$ of knots, where $H$ is the helicity of a given fluid and $t$ a formal constant. For oriented knotted vortex lines, $t^{H}$ satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, $t^{H}$ satisfies the skein relations of the Kauffman bracket polynomial. Our new algebraic method is to us…
It has been known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot K and any natural number n there exists a quasipositive knot Q whose Vassiliev invariants of order less than or equal to n coincide with those of K.
A lower bound of the crossing number of composite knots
针对纽结理论中复合纽结交叉数的老问题，本文证明了非平凡复合纽结交叉数的一个下界，该下界改进了Lackenby的相关结果。
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.
We introduce \textit{dual graph diagrams} representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call \textit{biquasiles} whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.
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.
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.
For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover the equality holds if and only if D is an alternating diagram.
In this note, I give a method to construct rational Seifert surface for those smooth or piece-wise linear oriented knots in Lens space. I assume that the oriented knot has a regular projection on Heegaard torus and then construct rational Seifert surface on twist toroidal diagram.
An Analogue of Milnor's Invariants for Knots in 3-Manifolds
本文将Milnor不变量推广到闭有向3-流形中的纽结，提出Dwyer数，给出其在$S^1 \times S^2$连通和中的计算方法，并证明该不变量可检测一类不与平凡纽结同痕的纽结。
This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is formulated in terms of the biquandle, a generalization of the quandle of David Joyce (which, in turn, is a generalization of the fundamental group of the knot or link). We then introduce the concept of a bi-oriented quantum algebra which provides an algebraic context for the associated quantum invariant.
We study factorizations of HOMFLY polynomials of certain knots and oriented links. We begin with a computer analysis of knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs.
In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in the real space is extended to the case of nonoriented knots and links in the projective space. It is also shown that this notion cannot be extended to oriented links.
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