We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie algebra G_2. It is therefore related to G_2 in the same way that the HOMFLY polynomial is related to A_n and the Kauffman polynomial is related to B_n, C_n, and D_n. We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions.
In this paper we introduce two theories of finite type invariants for framed links with fixed linking matrix. We show that these thepries are related to the theory of Vassiliev invariants of framed links. We also study the corresponding spaces of ``chord diagrams''.
We prove that for knots and links that are closures of 5-braids, single-colored ADO-invariant of third order coincides with Links-Gould polynomial with specific choice of variables. We also conjecture that this is true for any knots and links and provide some arguments for that.
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 study Coxeter racks over $\mathbb{Z}_n$ and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants.
Minor typographical errors fixed. Cochran constructed many links with Alexander module that of the unlink and some nonvanishing Milnor invariants, using as input commutators in a free group and as an invariant the longitudes of the links. We present a different and conjecturally complete construction, that uses elementary properties of clasper surgery, and a different invariant, the tree-part of the LMO invariant. Our method also constructs links with trivial higher Alexander modules and nontrivial Milnor invariants.
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
We extend our previous work from arXiv:1903.06863 on biquandle module invariants of oriented surface-links to the case of unoriented surface-links using bikei modules. The resulting infinite family of enhanced invariants proves be effective at distinguishing unoriented and especially non-orientable surface-links; in particular, we show that these invariants are more effective than the bikei homset cardinality invariant alone at distinguishing non-orientable surface-links. Moreover, as another application we note that our previous biquandle modules which do not satisfy the bikei module axioms a…
We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.
Using basic topology and linear algebra, we define a plethora of invariants of boundary links whose values are power series with noncommuting variables. These turn out to be useful and elementary reformulations of an invariant originally defined by M. Farber.
We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.
In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link invariant. This new invariant can be thought of as a multivariable version of the Links-Gould invariant. We also show that after a variable reduction our invariant specializes to the Conway potential function, which is a version of the multivariable Alexander polynomial.
We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N=1 are biquandles, which we call bikei. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, and we give an example of a non-involutory birack whose counting invariant detects the non-invertibility of a virtual knot.
Pulling back the weight system associated with the exceptional Lie algebra G_2 by a modification of the universal Vassiliev-Kontsevich invariant yields a link invariant; extending it to 3-nets, we derive a recursive algorithm for its evaluation.
We introduce an associative algebra Z[X,S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of Z[X,S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by the Alexander polynomial or the unenhanced birack counting invariants.
In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and gen…
M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum $(sl_n,\land V_n)$ link invariant, where $\land V_n$ is the set of the fundamental representations of the quantum group of $sl_n$. In the case of a [1,k]-colored link diagram, we prove that its homology is a link invariant. In the case of an [i,j]-colored link diagram, we define a normalized Poincare polynomial of its homology and prove the polynomial is a link invariant.
On the asymptotic expansion of quantum invariants related to surgeries of Whitehead link I: Relative Reshetikhin-Turaev invariants and the Turaev-Viro invariants at $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{2}}}$
This paper generalize [7](math.GT/0601291): We construct new links invariants from g, a type I basic classical Lie superalgebra. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module. Using this, we get a multivariable link invariant associated to any one parameter family of irreducible g-modules.
Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators. At present two examples of generalized Yang-Baxter operators are known and recently three types of variations for one of these were discovered. We present the definition of enhanced generalized YB-operators and show that all known examples of generalized YB-operators can be enhanced to give corresponding invariants of oriented links. Most of these invariants are specializations of the polynomial invariant $P$. Invariants from generalized YB-operators ar…
All polynomial invariants of links for two dimensional solutions of Yang-Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's $G_2$ quantum invariants. Our method consists in the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.
The usual construction of link invariants from quantum groups applied to the superalgebra D_{2 1,alpha} is shown to be trivial. One can modify this construction to get a two variable invariant. Unusually, this invariant is additive with respect to connected sum or disjoint union. This invariant contains an infinity of Vassiliev invariants that are not seen by the quantum invariants coming from Lie algebras (so neither by the colored HOMFLY-PT nor by the colored Kauffman polynomials).
Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show these Hopf algebras share properties with the $-1$ specialization of $U_q gl(n|1)$ leading to the proof of a conjecture of David De Wit, Atsushi Ishii and Jon Links on the Links-Gould invariants.
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that, for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the …
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.
We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma--Hecke algebras. The isomorphism relates affine Yokonuma--Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma--Hecke algebras, and in turn, to produce invariants for links in the solid torus. By restriction, this construction contains the construction of invariants for classical links from classical Yokonuma--Hecke algebras. In general, the obtained invariants form an infinite family of 3-variables polynomials. As a consequence…
Virtual links were introduced by Kauffman in 1999. We characterize the virtual link invariants that are partition functions of vertex models (as considered by de la Harpe and Jones), both in the real and in the complex case. We show that for any fixed number of states, these invariants form an affine variety. Basic techniques are the first and second fundamental theorem of invariant theory for the orthogonal group (in the sense of Weyl) and some related methods from algebraic geometry.
We present state sums for quantum link invariants arising from the representation theory of $U_q(\mathfrak{gl}_{N|M})$. We investigate the case of the $N$-th exterior power of the standard representation of $U_q(\mathfrak{gl}_{N|1})$ and explicit the relation with Kashaev invariants.
Costantino--Geer--Patureau-Mirand proved relations between the Reshetikhin--Turaev link invariants and the re-normalized link invariants for knots. Their theorem says that residues of the re-normalized link invariants are given by the Reshetikhin--Turaev link invariants. However, in the case of links, the residues are vanish, so we can not obtain any relations from the residues. In this paper, we prove that the Reshetikhin--Turaev link invariants appear in higher order terms of the re-normalized link invariants for links which are plumbed.
We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.
We show that a variation of Milnor's $\barμ$-invariants, the so-called Campbell-Hausdorff invariants introduced recently by Stefan Papadima, are of finite type with respect to {\it marked singular links}. These link invariants are stronger than quantum invariants in the sense that they detect easily the non-invertibility of links with more than one components. It is still open whether some effectively computable knot invariants, e.g. finite type knot invariants (Vassiliev invariants), could detect the non-invertibility of knots.
Pulling back the weight systems associated with the exceptional Lie algebras and their standard representations by a modification of the universal Vassiliev-Kontsevich invariant yields link invariants; extending them to coloured 3-nets, we derive for each of them a skein relation.
We discuss multivariable invariants of colored links associated with the $N$-dimensional root of unity representation of the quantum group. The invariants for $N>2$ are generalizations of the multi-variable Alexander polynomial. The invariants vanish for disconnected links. We review the definition of the invariants through (1,1)-tangles. When $(N,3)=1$ and $N$ is odd, the invariant does not vanish for the parallel link (cable) of the knot $3_1$, while the Alexander polynomial vanishes for the cable link.
Topological interpretation of the link invariants associated with the Weinstein--Xu classical solutions of the quantum Yang-Baxter equation are provided.
We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY-PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the gl(m|n) quantum invariants) as link invariants with values in the quantized oriented Brauer category.
Using the quantum double method, we construct a universal R-matrix for the quantization of the Lie superalgebra D(2,1,x). We use this result to construct a link invariant and show it coincides with a specialization of Kauffman's Dubrovnik polynomial.
Kashaev and Reshetikhin proposed a generalization of the Reshetikhin-Turaev link invariant construction to tangles with a flat connection in a principal G-bundle over the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to sl2, defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defi…
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.
The link invariant, arising from the cyclic quantum dilogarithm via the particular $R$-matrix construction, is proved to coincide with the invariant of triangulated links in $S^3$ introduced in R.M. Kashaev, Mod. Phys. Lett. A, Vol.9 No.40 (1994) 3757. The obtained invariant, like Alexander-Conway polynomial, vanishes on disjoint union of links. The $R$-matrix can be considered as the cyclic analog of the universal $R$-matrix associated with $U_q(sl(2))$ algebra.
We show that multivariable colored link invariants are derived from the roots of unity representations of $U_q(g)$. We propose a property of the Clebsch-Gordan coefficients of $U_q(g)$, which is important for defining the invariants of colored links. For $U_q(sl_2) we explicitly prove the property, and then construct invariants of colored links and colored ribbon graphs, which generalize the multivariable Alexander polynomial.
We have new solutions to the Yang-Baxter equation, from which we have constructed new link invariants containing more than two arbitrary parameters. This may be regarded as a generalization of the Jones' polynomial. We have also found another simpler invariant which discriminates only the linking structure of knots with each other, but not details of individual knot.
In this paper, we show that Hennings construction of invariants of framed links and 3-manifolds obtained from Hopf algebras can also be carried out for some algebraic quantum groups.
The most general R-matrix type state sum model for link invariants is constructed. It contains in itself all R-matrix invariants and is a generating function for "universal" Vassiliev link invariants. This expression is more simple than Kontsevich's expression for the same quantity, because it is defined combinatorially and does not contain any integrals, except for an expression for "the universal Drinfeld's associator".
We define and study the properties of observables associated to any link in $Σ\times {\bf R}$ (where $Σ$ is a compact surface) using the combinatorial quantization of hamiltonian Chern-Simons theory. These observables are traces of holonomies in a non commutative Yang-Mills theory where the gauge symmetry is ensured by a quantum group. We show that these observables are link invariants taking values in a non commutative algebra, the so called Moduli Algebra. When $Σ=S^2$ these link invariants are pure numbers and are equal to Reshetikhin-Turaev link invariants.
This paper defines a new sequence of finite dimensional algebras as quotients of the group algebras of the braid groups. This sequence depends on three homogeneous parameters and has a one-parameter family of Markov traces, and so gives a three parameter invariant of oriented links.
We present a mathematically clean review of our previous results on 1/K expansion of the colored Jones polynomial and on perturbative invariants of 3d rational homology spheres. We also prove that perturbative invariants defined through the stationary phase surgery formula are invariant under Kirby moves.
We explain how existing results (such as categorical sl(n) actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the HOMFLY polynomial of links coloured by arbitrary partitions. The construction uses a categorified HOMFLY clasp defined via cabling and infinite twists. We briefly discuss differentials and speculate on related structures.
The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V_infinity containing elements g_i satisfying the usual braid group relations and elements a_i satisfying g_i - g_i^{-1} = epsilon a_i, where epsilon is a formal variable that may be regarded as measuring the failure of g_i^2 to equal 1. Topologically, the elements a_i signify crossings. We show that a large class of link invariants of finite type are in one-to-one corres…
We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. This shows that the conjectural links-quivers correspondence of Kucharski-Reineke-Stošić-Sułkowski as well as the LMOV conjecture hold for rational links. Along the way, we extend the links-quivers correspondence to tangles and, thus, explore elements of a skein theory for motivic Donaldson-Thomas invariants.
We introduce an associative glueing operation $\oplus_q$ on the space of solutions of the Quantum Yang-Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves $2\times 2$ quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard $su(n)$ solution, its associated quantum matrix gr…
There is a known connection between the osp(1|2n) polynomial knot invariant $J_K^n$ and the so(2n+1) knot invariant ${}_{so} J_K^n$ studied by Clark in arXiv:1509.03533 and Blumen in arXiv:0901.3232. In the rank one case, the uncolored $U_{q}(osp(1|2))$ link invariant is equal to the $U_{t^{-1}q}(sl_2)$ link invariant where $t^2=-1$. We define a skein relation similar to the Kauffman bracket, and use that to recover an oriented link invariant which coincides with Clark's uncolored osp(1|2)-link invariant. This definition also comes from the representation theory of $U_{q,π}(sl_2)$, but using d…
We give a generalization of the Reshetikhin-Turaev functor for tangles to get a combinatorial formula for the universal Vassiliev-Kontsevich invariant of framed oriented links which is coincident with the Kontsevich integral. The universal Vassiliev-Kontsevich invariant is constructed using the Drinfeld associator. We prove the uniqueness of the Drinfeld associator. As a corollary one gets the rationality of the Kontsevich integral. Many properties of the universal Vassiliev-Kontsevich invariant are established. Connections to quantum group invariants and to multiple zeta values are discussed.
We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which is a specialization of joining virtual link diagram tangles. Basic techniques are the first fundamental theorem of invariant theory, the Hanlon-Wales theorem on the decomposition of Brauer algebras, and the Procesi-Schwarz theorem on inequalities for closed orbits.
In this paper we indicate one method of construction of linear representations of groups and algebras with translation invariant (except, maybe , finite number) defining relationships. As an illustration of this method, we give one approach to the construction of linear representations of braid group and derive some series of such representations. Some invariants of oriented knots and links are constructed. The author is grateful to Yuri Drozd, Sergey Ovsienko and other members of algebraic seminar at Kiev University for the creative atmosphere without which this work could hardly appear.
The occurrence of the Askey-Wilson (AW) algebra in the $SU(2)$ Chern-Simons (CS) theory and in the Reshetikhin-Turaev (RT) link invariant construction with quantum algebra $U_q(\mathfrak{su}_2)$ is explored. Tangle diagrams with three strands with some of them enclosed in a spin-$1/2$ closed loop are associated to the generators of the AW algebra. It is shown in both the CS theory and RT construction that the link invariant of these tangles obey the relations of the AW generators. It follows that the expectation values of certain Wilson loops in the CS theory satisfy relations dictated by the …
