Jérôme Dubois, Igor G. Korepanov, Evgeniy V. Martyushev
We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory.
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We modify the definition of spherical knotoids to include a framing, in analogy to framed knots, and define a further modification that includes a secondary 'coframing' to obtain 'biframed' knotoids. We exhibit topological spaces whose ambient isotopy classes are in one-to-one correspondence with framed and biframed knotoids respectively. We then show how framed and biframed knotoids allow us to generalize quantum knot invariants to a knotoid setting, leading to the construction of general Reshetikhin-Turaev type biframed knotoid invariants.
This article gives a foundational account of various characterizations of framed links in the $3$-sphere.
We introduce BV-algebra structures on the homology of the space of framed long knots in $\mathbb{R}^n$ in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The latter can be applied to Bousfield-Salvatore spectral sequence converging to the homology of the space of framed long knots. Conjecturally these two structures coincide with each other.
We consider the following question: when is the manifold obtained by gluing together two knot complements an $L$-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an $L$-space. We extend this result to allow for arbitrary integer framings. We find that splicing two integer framed nontrivial knot complements only produces an $L$-space if both knots are $L$-space knots and the framings lie in an appropriate range. The proof involves a careful analysis of the bordered Heegaard Floer invariants of each knot complement.
We generalize Ng's two-variable algebraic/combinatorial $0$-th framed knot contact homology for framed oriented knots in $S^3$ to knots in $S^1 \times S^2$, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ng's three-variable knot contact homology. Our main tool is Lin's generalization of the Markov theorem for braids in $S^3$ to braids in $S^1 \times S^2$. We conjecture that our framed cord algebras are always finitely generated for non-local kno…
A. Henrich proved the existence of the universal finite-type invariant of order one for virtual knots. We extend the construction and the methods of her paper to framed virtual knots. To do so, we introduce the notions of virtual strings and based matrices for framed flat virtual knots.
Framed holonomic knots
完整纽结是三维空间中的一个纽结，它是圆上一个光滑函数的2-喷注扩张。一个完整的纽结关联到一个通用的功能，自然是由黑板框架的纽结图相关联的1-喷流扩展的功能。框架纽结图有两个经典的不变量：Whitney指数（旋转数）W和自环绕数S。对于一个框架完整纽结，我们证明了W是有界的负的辫子指数的纽结，和W和|S|的总和是有界的负的欧拉示性数的任何塞弗特表面的纽结。本文证明了W=m的框架完整纽结的不变量S分裂成n个，其中n是最大的自然数，2n < |m| +1是整数不变量.利用这一点，框架完整纽结的框架完整合痕分类比它们的图的正规合痕分类更精细。
On Hecke lifting conjecture for framed knots
本文证明了框架纽结的Hecke提升猜想：对环面纽结给出完整证明，并对任意框架纽结验证其极限形式，该猜想源于Chern-Simons量子不变量的整性结构。
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 …
Vassiliev invariants of Legendrian, of transverse and framed knots in contact 3-manifolds
我们表明，对于一个大类的接触3-流形的Vassiliev不变量的Legendrian和框架结的群体是规范同构的。作为推论，我们得到曲面$F$上波前的有穷级Arnold $J^+$型不变量群同构于$F$的球面余切丛$ST^*F$中框架纽结的Vassiliev不变量群.另一方面，我们构建接触流形的第一个例子，其中勒让德结的Vassiliev不变量可以区分勒让德结，实现同位素框架结和勒让德浸入同伦。
We generalize three invariants, first discovered by A. Henrich, to the long and/or framed virtual knot case. These invariants are all finite-type invariants of order one, and include a universal one. The generalization will require us to extend the notion of a based matrix of a virtual string, first introduced by V. Turaev and later generalized by Henrich, to the long and framed cases.
Framed knot contact homology
本文将纽结接触同调推广到环$\mathbb{Z}[λ^{\pm 1},μ^{\pm 1}]$上的一个理论，给出了拓扑和组合的不变量。改进的不变量定义在$S^3$中的框架纽结上，并可推广到任意流形上的纽结，它能区分非纽结和突变体。它包含亚历山大多项式，并自然产生一个二元多项式结不变量，这是相关的$A$ -多项式。
Knot cobordism, torsion order and framed instanton homology
本文为瞬子纽结同调构造配边映射，恢复挠阶不等式，并据此计算交错纽结Dehn手术的带框瞬子Floer同调。
Cubes, cacti, and framed long knots
该研究定义了射影无刺仙人掌operad对标架长纽结空间泰勒塔各层的作用，证明其与Budney小2-立方体作用的空间相容性，改进了相关Vassiliev不变量猜想的前期结果，还重新证明了Sakai发现的一类Browder括号类的非平凡性。
We introduce and study the writhe of a permutation, a circular variant of the well-known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non-Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model. The distribution of the framing in this model is equivalent to the writhe of random permutations.
The universal order one invariant of framed knots in most S^1-bundles over orientable surfaces
众所周知，自连接是S^3中框架纽结唯一的Z值瓦西里耶夫不变量。然而，对于大多数3-流形，特别是对于可定向曲面F（而不是S^2）上的S^1-丛的全空间，Z值一阶不变量空间是无穷维的。本文给出了可定向的非紧曲面F（非S^2）上S^1-丛的可定向全空间中框架纽结的一阶不变量I的一个显式公式。我们证明了如果F不是S^2或S^1 × S^1，则I是泛一阶不变量，即它区分每两个可以由值在阿贝尔群中的一阶不变量区分的框架纽结.
BPS invariants from framed links
本文将代数曲线与纽结BPS不变量的关系推广到带标架纽结，给出相关显式公式，并对带标架链环的相关不变量进行数值计算验证其整性。
We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves. We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under Ω3 moves depends only on the forward/backward character of the move, similar to how the change of …
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.
Birack Bracket Quivers and Framed Links
本文提出birack括号和quiver值不变量，用于研究framed结和链的多项式不变量。
Framed instanton homology of surgeries on L-space knots
本文计算了具有瞬子L-空间手术的纽结在所有整数手术上的带标架瞬子Floer同调，并证明若纽结同时具有Heegaard Floer和瞬子Floer的L-空间手术，则两种理论在所有整数手术上一致。
Framed knots in 3-manifolds and affine self-linking numbers
对应于给定的非框架结$K\subset S^3$的非同位素框架结的数目$|K|$是无限的。这是由于零同调框架结的自连接数$\slk$的存在.我们使用Vassiliev-Goussarov不变量的方法来构造“仿射自环绕数”，它们是$\slk$到非零同调框架结的扩展。作为一个推论，我们得到，$|K|=\infty$的所有节点在一个定向（不一定是紧的）3-流形$M$，是不能实现为一个连接和$(S^1\times S^2) # M'$。紧致流形的这个结果首先由Hoste和Przytycki提出。他们提到的作品麦卡洛的想法的证明，但据我们所知，证明这一基本事实是没有文学。我们的证明是基于不同的想法。对于$M=(S^1\times S^2) # M'$，我们在$M$中构建$K$，使得$|K|=2\neq \infty$ .
Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-sectional inflation. Our main result gives explicit parameterizations for all periodic critical framed curves for this generalized functional. The main technical tool is a correspondence between the moduli space of shape similarity classes of closed framed curves and an infinite-dimen…
The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries are rational homology cobordant 3-manifolds, wherein the knots are not rationally concordant (that is not concordant in any rational homology S^3 x [0,1]). Specifically, we prove that, for any positive integer p and any knot K, the zero framed surgery on K is Z[1/p]-homology cob…
We define two concordance invariants of knots using framed instanton homology. These invariants $ν^\sharp$ and $τ^\sharp$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $τ$ invariant in Heegaard Floer homology. We use $ν^\sharp$ and $τ^\sharp$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyper…
We produce a facial state sum on plane diagrams of a knot or a link which admits an invariant specialization under Polyak's recent set of generating of 4 Reidemeister moves. Thus an isotopy invariant of framed links is obtained. Each state is a complete coloring of the faces of the diagram into white and black faces so that no two black faces share an edge. Each state induces a monomial in a ring of 16 variables. The sum of the states, properly specialized defines the new invariant. In despite of its simplicity it complements Jones invariant in distinguishing mirror pairs of links. In particul…
The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007), we explore further the structures of the $p$-adic framed braids and the $p$-adic Yokonuma-Hecke algebras constructed in Juyumaya and Lambropoulou (2007), by means of dense sub-structures approximating $p$-adic elements. We also construct a $p$-adic Markov trace on the $p$-adic Yokonuma-Hecke algebras and we approximate the values of the $p$-adic trace on $p$-adic elements. Surprisingly, the Markov traces do not re-scale …
Finite Order Invariants of Legendrian, Transverse, and Framed Knots in Contact 3-manifolds
我们证明了对于一个大类的接触流形的顺序$\leq n$不变量（值在任意阿贝尔群）的勒让德，横向和框架结的群体是典型同构。另一方面，对于$S^1\times S^2$上的任意共定向切触结构，其接触丛的非零欧拉类，我们构造了勒让德同伦勒让德结$K_1$和$K_2$的例子，使得它们实现同位素框架结，但可以通过$S^1\times S^2$中勒让德结的有穷级不变量来区分。我们构造了一大类接触流形$M$的类似例子，使得$M$是一个不可定向曲面上的局部平凡$S^1$ -纤维化的全空间。我们证明了在某些例子中$K_1$和$K_2$的补集是过扭曲的。
This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen's conjecture that a spectral sequence relates the two knot homologies. We replace the language of twisted coefficients with the more quantum topological language of framings on trivalent graphs. We define a homology theory for framed trivalent graphs with boundary that -- for a …
There are non homotopic framed homotopies of long knots
设$\mathcal {M}$是3-空间中所有的空间，包括奇异的长纽结，并且对于该空间，到平面的固定投影是浸入。设$cl(Σ^{(1)}_{iness})$是$\mathcal {M}$中所有奇异结点的并集的闭包，其中恰好有一个普通的二重点，并且使得两个分解表示相同的（非奇异的）结点类型。我们称$Σ^{(1)}_{iness}$为{ \em非本质墙}，称$\mathcal {M}_{ess} = \mathcal {M} \setminus cl(Σ^{(1)}_{iness})$为{ \em本质图空间}。通过对Kauffman括号的扩展，构造了$H^1(\mathcal {M}_{ess}; \mathbb{Z}[A, A^{-1}])$中的一个非平凡类.这特别意味着在$\mathcal {M}_{ess}$中有环，它由结的规则同位素和交叉变化组成，并且在$\mathcal {M}_{ess}$中是不可收缩的（导致论文的标题）。我们猜想，我们的建设产生了一个新的结多项式的unknotting号码1。
We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau …
For a knot $K,$ a slope $r$ is said to be characterizing if for no other knot $J$ does $r$-framed surgery along $J$ yield the same manifold as $r$-framed surgery on $K.$ Applying a condition of Baker and Motegi, we show that the knots $T_{2,2n+3}\#T_{-2,2n+1}$ have infinitely many non-characterizing slopes.
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.
Isomorphism of the groups of Vassiliev Invariants of Legendrian and of Pseudo Legendrian Knots in Contact 3-manifolds
对勒让德纽结的Vassiliev不变量的研究始于D. Fuchs和S. Tabachnikov谁表明，集团的复值瓦西里耶夫不变量的Legendrian和框架结的标准接触$R^3$是典型的同构。最近，我们构建了第一个例子，其中Vassiliev不变量的Legendrian和框架结是不同的，和Vassiliev不变量的Legendrian结区分Legendrian结是同位素的框架结和同伦的Legendrian浸入。这就提出了一个问题：关于勒让德结的什么信息可以用瓦西里耶夫不变量来捕捉。在这里，我们回答这个问题，表明任何接触3-流形的cooriented的切触结构的勒让德结和结的瓦西里耶夫不变量的群体是无处正切的向量场coorients的切触结构是规范同构的。
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knot…
Singularities of translation surfaces under the linearly dependent condition in Euclidean 3-space
本文研究了在欧几里得3空间中，线性依赖条件下翻译表面的奇点特性，通过广义框架表面理论分析奇点类型。
Framed instanton homology and Frøyshov's invariant
本文研究了结的Dehn手术中框架瞬子同调与Froyshov不变量的关系，并证明了非退化$SU(2)$-abelian的限制条件。
We generalize H. Seifert's algorithm for finding a Seifert surface for a knot or link. The generalization applies to "framed oriented measured lamination links." For knots, a Seifert surface determines a unique framing. In our setting, we analyze the set of framed lamination links which bound Seifert laminations and are carried by an $S^1$-fibered tube neighborhood of an oriented train track embedded in a 3-manifold.
Maps from knots to $2$-links, chord diagrams, and a way to enhance Vassiliev invariants
研究如何将瓦西里耶夫纽结不变量等推广到特定带框弦图，核心方法是探讨相关映射，主要贡献是提供推广方式。
We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These invariants are invariants of concordance and of J-characteristicness when f induces the identity on the fundamental group of M, and may be viewed as extensions of Milnor's invariants. We do not require the knots considered here to be rationally null-homologous or framed.
Framed instanton homology and concordance, II
本文研究了框架瞬子同调与结的同调，证明了$ν^\sharp(K)$的性质，并定义了瞬子FloER的类似量$ε^\sharp(K)$，同时探讨了3-流形作为结手术的问题。
Generalized rational blow-down, torus knots, and Euclidean algorithm
我们构造了Jongil Park提出的“广义有理blow-down”中所使用的有理同调球的Kirby图。该图由一个虚线圆和一个环面结组成。链接更简单，但参数有点复杂。在构造和证明过程中，三次使用了欧几里德算法。
The hyperbolic structure on a 3-dimensional cone-manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number. This stands in contrast to hyperbolic volumes whose number-theoretic nature is usually quite complicated.
Euclidean tetrahedra and knot invariants
我们构造纽结不变量的基础是将欧几里得几何值赋予纽结所在的球面S^3的三角剖分。主要的新特点，这种建设相比，作者的早期论文流形不变量是，现在非零的“赤字角”（术语的Regge演算）也可以处理。此外，节点精确地沿着具有非零亏损角的三角剖分的那些边。
Knot surgery formulae for instanton Floer homology II: applications
本文通过整数手术公式研究了3流形的瞬子同调，揭示了几乎L空间结的纤维化和强准正性质，并确定了特定类型的结结构。
In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the $\mathfrak{gl}_N$ Reshetikhin--Turaev link invariants and the universal $\mathfrak{gl}_N$ knot invariant; 2) an explicit construction of the unified $\mathfrak{gl}_N$ invariants for integral homology 3-spheres using universal Kirby colors. These results …
This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a $3$-manifold or a null-homologous knot inside a $3$-manifold and the Heegaard diagram of that $3$-manifold or knot. We further use this relation to compute the instanton knot homology of some families of $(1,1)$-knots, including all torus knots in $S^3$, which were mostly unknown before. As a second application, we also study the relation between the instanto…
We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not binary dihedral.
We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.
We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the $n$-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres from a slice knot with unknotting number one.
Virtual Knot Diagrams and the Witten-Reshetikhin-Turaev Invariant
将经典链图的Witten-Reshetikhin-Turaev不变量推广到虚链图。这个不变量不受框架Reidemeister移动和Kirby演算的影响。因此，它也是由经典链接图表示的3-流形的不变量。这个推广被用来证明存在具有非平凡Witten-Reshetikhin-Turaev不变量和平凡3-流形基本群的虚纽结图。
Knot invariants from XC-structures on the Sweedler algebra are trivial
本文研究了从Sweedler代数上的XC结构得到的结不变量，证明其由结的框架决定，并展示了非源于带环霍普夫代数的XC结构示例。
We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures. To produce these examples, we introduce a new description of cork twists and utilize satellite maps. As an application, we produce knots with the same 0-surgery which are not concordant for any orientations, disproving the Akbulut-Kirby conjecture given in 1978.
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.
Singularities of helicoidal surfaces of frontals in the Euclidean space
本文研究欧氏空间中由含奇点前曲线生成的螺旋面，将其刻画为广义标架基曲面，并利用 Legendre 曲线曲率给出其不变量、曲率及奇点判据。
3-braids with 4-valent vertices and knot invariant
利用群$G_n^3$和辫子的plat闭包构造经典纽结的图值不变量，通过修改群$G_n^3$定义从辫子群到带叶子的6价图的映射，得到图等价类模局部移动的纽结不变量。
Embedding and knotting of manifolds in Euclidean spaces
对高维对象拓扑结构的清晰理解在纯数学和应用数学的许多分支中都很重要。在这个调查中，我们试图提出一些结果的高维拓扑结构的方式，使清晰的视觉和直观的部分的建设和论点。特别是，我们展示了如何抽象的代数结构自然出现在几何问题的研究。在给出一个一般的结构之前，我们用简单但重要的特殊情况来说明主要思想，在这些情况下，本质并不被技术细节所掩盖。更具体地说，我们提出了几个经典和现代的结果嵌入和打结的流形在欧几里得空间。我们陈述了许多具体的结果（特别是，最近明确的分类打结环面）。他们的声明（但不是证据!）对非专业人士来说是简单易行的。我们通过经典的van Kampen-Shapiro-Wu-Haefliger-Weber“删除产品”阻塞概述了嵌入的一般方法。这种方法将嵌入的合痕分类归结为等变映射的同伦分类，从而得到了上述的具体结果。我们描述了复兴的兴趣在这个美丽的分支拓扑结构，提出新的结果在这方面（弗里德曼，Krushkal，Teichner，西格尔，Spiez和作者）：一个推广的Haefliger-韦伯嵌入定理低于亚稳维范围和例子表明，其他类似物这个定理是假的亚稳维范围之外。
Turaev's shadow formula calculates the SU(2)-Reshetikhin-Turaev-Witten invariants using shadows, and its expression is somehow similar to a Euler characteristic. We give a short proof of this formula using skein theory. The formula applies to pairs (M,G) where M is a closed oriented 3-manifold and GcM is a (possibly empty) colored framed trivalent graph (for instance, a framed knot or link).
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…
We give a formula for the duality structure of the 3-manifold obtained by doing zero-framed surgery along a knot in the 3-sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blanchfield pairing of such 3-manifolds. With the twisting defined by Casson-Gordon style representations, we use our computation of the twisted Blanchfield pairing to show that some subtle satellites of genus two ribbon knots yield non-slice knots. The construction is subtle in the sense that, once based, the infection curve lies in the second derived s…
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