15 pages, 1 Table, submitted for publication in Progress in Mathematics volume in honor of the 60th birthday of Kolya Reshetikhin. A revised version with a detailed proof of Theorem 1; two more references added
By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed groups, we illustrate the construction and the importance of consideration of long knots. Else, by using the restricted dual of algebras and Drinfeld's quantum double construction, we show that to any Hopf algebra $H$ with invertible antipode, one can associate a universal long knot invariant $Z_H(K)$ taking its values in the convolution algebra $((D(H))^o)^*$ of the restricted dual Hopf algebra $(D(H))^o$ of the quantum double $D(H)$ of $H$. That extends the known constructions of universal invariants previously considered mostly either in the case of finite dimensional Hopf algebras or by using some topological completions.
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.
We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.
A connect sum formula for the two variable series invariant of a complement of knot is proposed. We provide two kinds of numerical evidence for the proposed formula by examining various torus knots.
We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for this invariant.
We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle 2-cocycle ϕ^0 and a map ϕ^1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide examples to show that the parity cocycle invariants can distinguish virtual knots which are not distinguished by the corresponding non-parity invariants.
We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images.
We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which are analogous to those of the invariant Upsilon (which arises from knot Floer homology), and some which differ. We make some explicit computations and give some topological applications. Further to this, we define a concordance invariant from equivariant sl(n) knot cohomology which subsumes many known concordance invariants arising from quantum knot cohomolo…
We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about knots modulo the nth derived subgroups of the pure braid groups, and about knots modulo braid subgroups in general.
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic interpretations of quandle cocycles as deformations of extensions are also given. The proofs rely on colored knot diagrams.
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree at most n in the classical sense. A universal invariant of degree at most n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove…
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinning. We also demonstrate relations between cocycle invariants and Alexander matrices.
Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle co…
I follow Y. Yokota to explain how to obtain a tetrahedron decomposition of the complement of a hyperbolic knot and compare it with the asymptotic behavior of Kashaev's link invariant using the figure-eight knot as an example.
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation --- the axioms of which model the Reidemeister moves in the classical theory of knotted and linked curves in 3-space. Colorings of diagrams of knotted curves and surfaces by quandle ele…
We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.
Kashaev's invariants for a knot in a three sphere are generalized to invariants of a knot in a three manifold. A relation between the newly constructed invariants and the hyperbolic volume of the knot complement is observed for some knots in lens spaces.
We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot diagram. The definition uses decompositions of knot diagrams: to a collection of points on the line, we associate a differential graded algebra; to a partial knot diagram, we associate modules over the algebra. The knot invariant is obtained from these modules by an appropriate tensor product.
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.
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor product representations of Kac-Moody algebras and quantum groups, constructed a prequel to this paper. These categories are based on the pictorial approach of Khovanov and Lauda. In this paper, we show that these categories are related by functors corresponding to the braiding and (co…
A relation between the two-variable series knot invariant and the Akutus-Deguchi-Ohtsuki(ADO)-invariant was conjectured recently. We reinforce the conjecture by presenting explicit formulas and/or an algorithm for certain ADO-invariants of torus knots obtained from the series invariant of complement of a knot. Furthermore, one parameter deformation of ADO_3-polynomial of torus knots is provided.
We show that the quantum covering group associated to osp(1|2n) has an associated colored quantum knot invariant à la Reshetikhin-Turaev, which specializes to a quantum knot invariant for osp(1|2n), and to the usual quantum knot invariant for so(1+2n). We then show that these knot invariants are the same, up to a change of variables and a constant factor depending on the knot and weight.
This an article about some elementary geometric and combinatorial natures of various knot energies. A related "new" knot invariant -- the X-crossing number -- is introduced.
We calculated the rational Khovanov homology of some class of pretzel knots, by using the spectral sequence constructed by P. Turner. Moreover, we determined the Rasmussen's s-invariant of almost of pretzel knots with three pretzels.
We discuss different invariants of knots and links that depend on a primitive root of unity. We clarify the definitions of existing invariants with the Reshetikhin-Turaev method, present the generalization of ADO invariants to $\mathcal{U}_q(sl_N)$ and highlight the connections between different invariants.
We use categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.
A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the fi…
We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M_11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish.
The universal Vassiliev invariant from the perturbative Chern-Simons theory is actually a knot invariant without any correction term. The anomaly considered by Bott and Taubes is proved to be zero.
By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of $q$-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.
Using the Racah coefficients in our earlier paper arXiv:1107.3918, we explicitly write the Chern-Simons field theory invariants for many non-torus knot and links. Further, we have tabulated the reformulated invariants which agrees with the Ooguri-Vafa conjecture for knots and Labastida-Marino-Vafa conjecture for links.
Using elementary counting methods of weight systems for finite type invariants of knots and integral homology 3-spheres, in the spirit of [B-NG], we answer positively three questions raised in [Ga]. In particular, we exhibit a one-to-one map from the space of finite type invariants of integral homology 3-spheres to the space of finite type invariants of knots in $S^3$.
Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge group. From this expansion new numerical knot invariants are obtained. These knot invariants turn out to be of finite type (Vassiliev invariants), and to possess an integral representation. Using known results about Jones, HOMFLY, Kauffman and Akutsu-Wadati polynomial invariants these new knot invariants are computed up to type six for all prime knots up to…
We show that the number of homomorphisms from a knot group to a finite group $G$ cannot be a Vassiliev invariant, unless it is constant on the set of $(2,2p+1)$ torus knots. In several cases, such as when $G$ is a dihedral or symmetric group, this implies that the number of homomorphisms is not a Vassiliev invariant.
We construct gauge invariant operators for singular knots in the context of Chern-Simons gauge theory. These new operators provide polynomial invariants and Vassiliev invariants for singular knots. As an application we present the form of the Kontsevich integral for the case of singular knots.
An invariant of knots is constructed from an integral for geometric braids due to Kohno and Kontsevich. It takes values in a quotient by a certain ideal of the algebra generated by chord diagrams over the circle.
