V. Kurlin, V. Vershinin
Construction of a semigroup with 15 generators and 84 relations is given. The center of this semigroup is in one-to-one correspondence with the set of all isotopy classes of non-oriented singular knots (links with finitely many double intersections in general position) in three-dimensional space.
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Given any closed, connected, orientable $3$--manifold and integers $g\geq g(M), D > 0$, we show the existence of knots in $M$ whose genus $g$ bridge number is greater than $D$. These knots lie in a page of an open book decomposition of $M$, and the proof proceeds by examining the action of the map induced by the monodromy on the arc and curve complex of a page. A corollary is that there are Berge knots of arbitrarily large genus one bridge number.
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.
In this note, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, ξ) as the minimal genus of a page of an open book of M supporting the contact structure ξsuch that L sits on a page and the framing given by the contact structure and by the page agree. We show any null-homologous loose knot in an overtwisted contact structure has support genus zero. To prove this, we observe any topological knot or link in any 3-manifold M sits …
This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.
A singular knot is an immersed circle in $\mathbb R^{3}$ with finitely many transverse double points. The study of singular knots was initially motivated by the study of Vassiliev invariants. Namely, singular knots give rise to a decreasing filtration on the infinite dimensional vector space spanned by isotopy classes of knots: this is called the Vassiliev filtration, and the study of the corresponding associated graded space has lead to many insights in knot theory. The Vassiliev filtration has an alternative, more algebraic definition for many flavours of knot theory, for example braids and …
Given a biquandle $(X, S)$, a function $τ$ with certain compatibility and a pair of {\em non commutative cocyles} $f,h:X \times X\to G$ with values in a non necessarily commutative group $G$, we give an invariant for singular knots / links. Given $(X,S,τ)$, we also define a universal group $U_{nc}^{fh}(X)$ and universal functions governing all 2-cocycles in $X$, and exhibit examples of computations. When the target group is abelian, a notion of {\em abelian cocycle pair} is given and the "state sum" is defined for singular knots/links. Computations generalizing linking number for singular knot…
A perturbed Alexander polynomial for singular knots
本文引入奇异纽结与tangle的Alexander多项式版本，通过引入扰动显著增强其功能，并证明对于奇异长结该多项式与先前定义一致。
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.
In this short survey we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent enhancements to this invariant. These enhancements include a singquandle cocycle invariant and several polynomial invariants of singular knots obtained from the singquandle structure. We then explore psyquandles which can be thought of as generalizations of oriented signquandles, and review recent developments regarding invariants of singular knots obtained…
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial graphs and singular knots, that is obtained by replacing unknown crossings or vertices by rational tangles.
Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$ such that its links are the total spaces of the $n$-branched cyclic coverings over the corresponding knots. In this way we obtain as links of singularities, $3$-manifolds that does not appear in the complex case, such as hyperbolic $3$-manifolds or the Hantzsche-Wendt manifold.
Singular twisted links and singular twisted virtual braids
本文引入奇异扭曲纽结理论，证明了奇异扭曲链的Alexander定理与Markov定理，并定义了奇异扭曲虚拟辫的幺半群结构及其约化表示。
We generalize the colored Jones polynomial to $4$-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.
In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma--Hecke algebras ${\rm Y}_{d,n}(u)$ and the theory of singular braids. The Yokonuma--Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid $SB_n$ into the algebra ${\rm Y}_{d,n}(u)$. Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relat…
A partial classification of 3-dimensional clasp number two, genus two fibered knots
本文在假设条件下的情况下，对亏格为二、clasp数为二的纤维化结进行了部分分类。
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex $C_{F}(S)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is isomorphic to the HOMFLY-PT homology of $S$. Together with a naturality condition on the induced edge maps, this conjecture would prove the spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on $C_{F}(S)$, a recursion formula for HOMFLY-PT homology, and additional $sl_{n}$-like differentials on $C_{F}(S)$, we prove this …
The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of knots that are closed braids is shown to lead to a classification theorem for transversal knots that are closed 3-braids. A list is given of all low crossing examples of transversally non-simple knots that are closed 3-braids.
Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in the (n+2)-space. There are two inequivalent embeddings of an n-manifold in the (n+2)-space such that the projection of one of these in (n+1)-space has no double points and the projection…
Non collapse of the Sinha spectral sequence for knots in R^3
本文显式描述了R^3中长纽结的Sinha模2谱序列至第三页，发现非平凡微分，与有理情形不同。
Localization and the Floer homology of strongly invertible knots
该研究在定向强可逆纽结K的纽结Floer同调中建立两个谱序列，分别关联K与三维球面的Heegaard Floer同调、K的奇异纽结Floer同调与K的商纽结Floer同调。
We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial and the invariant of singular knots is an extension of an invariant of singular knots previously defined by S. Lambropoulou and the second author.
A knot \widetilde{K} \subset S^3 is q-periodic if there is a \mathbb Z_q-action preserving \widetilde{K} whose fixed set is an unknot U. The quotient of \widetilde{K} under the action is a second knot K. We construct equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's classical condition on the Alexander polynomials of periodic knots is a quick consequence of these diagrams. For \widetilde{K} a two-periodic knot, we show there is a spectral sequence whose E^1 page is \hat{\mathit{HFL}}(S^3,\widetilde{K}\cup U)\otimes V^{\otimes (2n-1)})\otimes \mathbb Z_2((θ)) and whos…
We construct a cubical CW-complex CK(M^3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M^3. We construct \bar{CK}(R^3) by attaching cells to CK(R^3) for every degenerate 1-singular and 2-singular knot, and we show that π_1(\bar{CK}(R^3))=1 and π_2(\bar{CK}(R^3))=Z. We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R^3 we show that v_2 uniquely generates H^2(CK,D), where D is the subcomplex of degenerate singular knots. More generally, we show that any Vassiliev invariant coming from the Conway polynomial…
We introduce algebraic structures known as psybrackets and use them to define invariants of pseudoknots and singular knots and links. Psybrackets are Niebrzydowski tribrackets with additional structure inspired by the Reidemeister moves for pseudoknots and singular knots. Examples and computations are provided.
Given a knot $K$ inside an integer homology sphere $Y$, the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into $SU(2)$ which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4-manifold $X$ with the integral homology of $S^{1}\times S^{3}$, and an embedded torus $T$ inside $X$ such that $H_{1}(T;\mathbb{Z})$ surjects onto $H_{1}(X;\…
Given a knot K in S^3, let Σ(K) be the double branched cover of S^3 over K. We show there is a spectral sequence whose E^1 page is (\hat{HFK}(Σ(K), K) \otimes V^{n-1}) \otimes \mathbb Z_2((q)), for V a \mathbb Z_2-vector space of dimension two, and whose E^{\infty} page is isomorphic to (\hat{HFK}(S^3, K) \otimes V^{n-1}) \otimes \mathbb Z_2((q)), as \mathbb Z_2((q))-modules. As a consequence, we deduce a rank inequality between the knot Floer homologies \hat{HFK}(Σ(K), K) and \hat{HFK}(S^3, K).
In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the nontriviality of the desingularization morphism.
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.
The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.
Instanton knot invariants with rational holonomy parameters and an application for torus knot groups
本文推广了 Daemi-Scaduto 的等变奇异瞬子 Floer 理论，研究带有理完整参数的瞬子纽结不变量，并确定了除有限多个参数外环面纽结的不可约奇异瞬子同调，为 Slice-Ribbon 猜想的环面纽结版本提供了新证据。
Let $\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(Σ^{(1)}_{iness})$ be the closure of the union of all singular knots in $\mathcal {M}$ with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call $Σ^{(1)}_{iness}$ the {\em inessential walls} and we call $\mathcal {M}_{ess} = \mathcal {M} \setminus cl(Σ^{(1)}_{iness})$ the {\em essential diagram space}. We construct a non trivial class in $H^1(\mathcal {M}_{ess}; \mathbb{Z}[A,…
Given a knot K in the 3-sphere, consider a singular disk bounded by K and the intersections of K with the interior of the disk. The absolute number of intersections, minimised over all choices of singular disk with a given algebraic number of intersections, defines the framing function of the knot. We show that the framing function is symmetric except at a finite number of points. The symmetry axis is a new knot invariant, called the natural framing of the knot. We calculate the natural framing of torus knots and some other knots, and discuss some of its properties and its relations to the sig…
Unoriented skein exact triangles in equivariant singular instanton Floer theory
本文将等变奇异瞬子Floer理论从纽结扩展到非零行列式链环，证明了无定向辫正合三角形，实现了Murasugi符号行为的范畴化，并揭示了L-space纽结与瞬子范畴化的潜在联系。
The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Flo…
Chern-Simons functional, singular instantons, and the four-dimensional clasp number
本文利用源自等变奇异瞬子理论并与陈-西蒙斯泛函相关的纽结不变量，肯定回答了四维 clasp 数与切片亏格之差可否任意大的问题，同时解决 Livingston 猜想，并研究奇异瞬子 Frøyshov 不变量及其在无向切片亏格、环面纽结表示和协边中的应用。
We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.
In this paper, we show that any topological knot or link in $S^1 \times S^2$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $S^1 \times S^2$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces $L(p,1)$.
Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orie…
We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.
We introduce shadow structures for singular knot theory. Precisely, we define \emph{two} invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.
Grid homology for singular links in lens space and a resolution cube
本文定义透镜空间中奇异链环的网格同调，构造正则链环 knot Floer 同调的分解立方体，并证明其与经典同调一致，推广了奇异纽结理论至透镜空间。
Equivariant aspects of singular instanton Floer homology
本文利用$SU(2)$奇异瞬子规范理论为整数同调3-球面中的纽结构造等变不变量，包括类似纽结Floer同调的三元群和Frøyshov型配边不变量，并借助ADHM描述刻画球面纽结的模空间。
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…
We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We also introduce singular semiquandles and virtual singular semiquandles which define invariants of flat singular virtual knots and links. As an application, we use semiquandle invariants to compare two Vassiliev invariants.
In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.
Unknotting number, ribbon concordance, and singular instantons
利用等变奇异瞬子弗洛尔理论，研究通过带状协边得到的切片纽结的解结操作，发现同号解结序列需含两种符号，还给出三维流形类似情况，为德恩手术数单调性提供证据。
Singular instanton homology of dual knots
本文研究了双结的奇异瞬子同调的维数公式，并探讨了其在拓扑学中的应用。
We extend to the long virtual knot case the constructions first presented by A. Henrich and later generalized by the author to the framed virtual knot case. These consist of three Vassiliev invariants of order one, including a universal one, as well as the notions of a based matrix and a singular based matrix and their relation to long flat virtual knots.
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.
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 …
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians…
We determine the Thurston's geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space $S^2$ and no more than three exceptional fibres, whose singular set, composed by fibres, has at most 3 components which can include exceptional or general fibres (the total number of exceptional and singular fibres is less or equal than three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery in a torus knot $K_{(r,…
On the invariance of the Dowlin spectral sequence
本文证明 Dowlin 构造的连接 Khovanov 同调与镜像纽结 Floer 同调的谱序列，其 $E_k$ 页在 $k\ge 3$ 时均为链环不变量。
Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, d>33, that possess exactly one singularity which has exactly one Puiseux pair (p;q). The realized triples (p,d,q) are expressed as successive even terms in the Fibonacci …
Ribbon numbers of 12-crossing knots
本文研究纽结的带状数，通过Alexander多项式集合计算12交叉纽结的带状数，并发现该方法对高亏格情形不再有效。
Embedding calculus and Vassiliev spectral sequence
本文利用 Thom 空间模型变体证明 Vassiliev 与 Sinha 谱序列在域系数下 $E_\infty$ 页同构，并推出 Vassiliev 序列在有理数域上 $E_1$ 页退化。
We enhance the psyquandle counting invariant for singular knots and pseudoknots using quivers analogously to quandle coloring quivers. This enables us to extend the in-degree polynomial invariants from quandle coloring quiver theory to the case of singular knots and pseudoknots. As a side effect we obtain biquandle coloring quivers and in-degree polynomial invariants for classical and virtual knots and links.
A non-singular connected algebraic curve $A$ in a simply connected algebraic surface $X$ can be knotted so that its homology class and the fundamental group of its complement in $X$ is preserved, provided $A$ is sufficiently complex (not too ``rigid''). For example, it is true if $A$ admits a degeneration to an irreducible curve $A_0$ having a unique singularity of the type $X_9$ (a non-degenerate quadriple point), or more complicated one, and $A.A>16$. This generalizes the previous result of the author which concerns the curves in $CP^2$ of degree $d>4$ (the old preprint is included as a part…
Goodwillie's cosimplicial model for the space of long knots and its applications
本研究推导了Goodwillie提出的余单纯空间与区间开集到紧生成弱Hausdorff空间的良好函子间的对应，借此算出长纽结嵌入函子Taylor塔对应纤维化塔的整系数Bousfield–Kan同伦谱序列第一页，并引入(i,n)-标记单位三价图给出d¹微分的组合解释。
We use topological methods to study various semicontinuity properties of spectra of singular points of plane algebraic curves and of polynomials in two variables at infinity. Using Seifert forms and the Tristram--Levine signatures of links, we reprove (in a slightly weaker version) a result obtained by Steenbrink and Varchenko on semicontinuity of spectrum at infinity. We also relate the spectrum at infinity of a polynomial with spectra of singular points of a chosen fiber.
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