Link Invariants Research Articles

Cocycles of G-Alexander biquandles and G-Alexander multiple conjugation biquandles

Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in 3-space. Cocycles of biquandles and multiple conjugation quandles can be used to construct state-sum type invariants of links and handlebody-links. In this paper we discuss cocycles of a certain class of biquandles and multiple conjugation biquandles, which we call G-Alexander biquandles and G-Alexander multiple conjugation biquandles, with a relationship with group cocycles. We give a method to obtain a (biquandle or multiple conjugation biquandle) cocycle from a group cocycle.

Multi-switches, representations of virtual braids, and invariants of virtual links

Multi-switches, representations of virtual braids, and invariants of virtual links

3D topological models and Heegaard splitting. II. Pontryagin duality and observables

In Paper I [F. Thuillier, “3D topological models and Heegaard splitting I: Partition function,” J. Math. Phys. 60, 32 (2019)], a construction of the smooth Deligne–Beilinson cohomology groups HDp(M) on a closed 3-manifold M represented by a Heegaard splitting XL ∪fXR was presented. Then, the partition functions of the U(1) Chern–Simons and BF Quantum field theories were determined from this construction. In this second and concluding article, we stay in the context of a Heegaard spitting of M to define Deligne–Beilinson 1-currents whose equivalent classes form the elements of HD1(M)⋆, the Pontryagin dual of HD1(M). Finally, we use singular fields to first recover the partition functions of the U(1) Chern–Simons and BF quantum field theories and next to determine the link invariants defined by these theories. The difference between the use of smooth and singular fields is also discussed.

Full colored HOMFLYPT invariants, composite invariants and congruence skein relations

In this paper, we investigate the properties of certain quantum invariants of links by using the HOMFLY skein theory. First, we obtain the limit behavior for the full colored HOMFLYPT invariant which is the natural generalization of the colored HOMFLYPT invariant. Then we focus on the composite invariant which is a certain combination of the full colored HOMFLYPT invariants. Motivated by the study of the Labastida–Marino–Ooguri–Vafa conjecture for the framed composite invariants of links, we introduce the notion of reformulated composite invariant $$\check{{\mathcal {R}}}_{p}({\mathcal {L}};q,a)$$ . By using the HOMFLY skein theory, we prove that $$\check{{\mathcal {R}}}_{p}({\mathcal {L}};q,a)$$ actually lies in the integral ring $$2{\mathbb {Z}}[(q-q^{-1})^2,a^{\pm 1}]$$ . Finally, we propose a conjectural congruence skein relation for $$\check{{\mathcal {R}}}_{p}({\mathcal {L}};q,a)$$ and prove it for certain special cases.

Multi-Switches, Representations of Virtual Braids, and Invariants of Virtual Links

Multi-Switches, Representations of Virtual Braids, and Invariants of Virtual Links

Leonard pairs, spin models, and distance-regular graphs

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard pair, said to have spin. The notion of a spin model was introduced by V.F.R. Jones to construct link invariants. A spin model is a symmetric matrix over C that satisfies two conditions, called the type II and type III conditions. It is known that a spin model W is contained in a certain finite-dimensional algebra N(W), called the Nomura algebra. It often happens that a spin model W satisfies W∈M⊆N(W), where M is the Bose-Mesner algebra of a distance-regular graph Γ; in this case we say that Γ affords W. If Γ affords a spin model, then each irreducible module for every Terwilliger algebra of Γ takes a certain form, recently described by Caughman, Curtin, Nomura, and Wolff. In the present paper we show that the converse is true; if each irreducible module for every Terwilliger algebra of Γ takes this form, then Γ affords a spin model. We explicitly construct this spin model when Γ has q-Racah type. The proof of our main result relies heavily on the theory of spin Leonard pairs.

Some remarks on the chord index

In this paper, we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots.

Transverse link invariants from the deformations of Khovanov 𝔰𝔩3–homology

In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $\beta_3$-invariants) which are transverse braid invariants. This family includes Wu's $\psi_{3}$-invariant. Furthermore, we analyse the vanishing of the homology classes of the $\beta_3$-invariants and relate it to the vanishing of Plamenevskaya's and Wu's invariants. Finally, we use the $\beta_3$-invariants to produce some Bennequin-type inequalities.

Higher arity self-distributive operations in Cascades and their cohomology

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive [Formula: see text]-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing [Formula: see text]-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.

Surgery on links of linking number zero and the Heegaard Floer $d$-invariant

We study Heegaard Floer homology and various related invariants (such as the h -function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the h -function, the Sato–Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer d -invariants of integral surgeries on two-component L-space links of linking number zero in terms of the h -function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the \nu^{+} -invariants of the knots obtained from blowing down the components. We give a proof of a skein inequality for the d -invariants of +1 surgeries along linking number zero links that differ by a crossing change. We also describe bounds on the smooth four-genus of links in terms of the h -function, expanding on previous work of the second author, and use these bounds to calculate the four-genus in several examples of links.

The C-complex clasp number of links

In the 1980s, Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then, this has been extended by works of Cimasoni, Florens, Mellor, Melvin, Conway, Toffoli, Friedl, and others to compute other link invariants. Informally, a C-complex is a union of surfaces which are allowed to intersect each other in clasps. We study the minimal number of clasps amongst all C-complexes bounded by a fixed link L. This measure of complexity is related to the number of crossing changes needed to reduce L to a boundary link. We prove that if L is a 2-component link with nonzero linking number, then the linking number determines the minimal number of clasps amongst all C-complexes. In the case of 3-component links, the triple linking number provides an additional lower bound on the number of clasps in a C-complex.

The structure of biquandle brackets

In their paper entitled “Quantum Enhancements and Biquandle Brackets”, Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We prove herein that if a biquandle bracket [Formula: see text] is the pointwise product of the pair of functions [Formula: see text] with a function [Formula: see text], then [Formula: see text] is also a biquandle bracket if and only if [Formula: see text] is a biquandle 2-cocycle (up to a constant multiple). As an application, we show that a new invariant introduced by Yang factors in this way, which allows us to show that the new invariant is in fact equivalent to the Jones polynomial on knots. Additionally, we provide a few new results about the structure of biquandle brackets and their relationship with biquandle 2-cocycles.

State sum invariants for flat virtual links from the chord index

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

Multivariate Alexander quandles, II. The involutory medial quandle of a link

Joyce showed that for a classical knot [Formula: see text], the order of the involutory medial quandle is [Formula: see text]. Generalizing Joyce’s result, we show that for a classical link [Formula: see text] of [Formula: see text] components, the order of the involutory medial quandle is [Formula: see text]. In particular, [Formula: see text] is infinite if and only if [Formula: see text]. We also relate [Formula: see text] to several other link invariants.

Holonomy braidings, biquandles and quantum invariants of links with $$\mathsf {SL} _2(\mathbb {C} )$$ flat connections

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev’s homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group Uqsl(2) at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin–Turaev ribbon type functor for tangles with quandle representations. This theory applies to the unrestricted quantum group Uqsl(2) and produces an invariant of links with a gauge class of quandle representations.

Two Parameters bt-Algebra and Invariants for Links and Tied Links

We introduce a two-parameters bt-algebra which, by specialization, becomes the one-parameter bt-algebra, introduced by the authors, as well as another one-parameter presentation of it; the invariant for links and tied links, associated to this two-parameter algebra via Jones recipe, contains as specializations the invariants obtained from these two presentations of the bt-algebra and then is more powerful than each of them. Also, a new non Homflypt polynomial invariant is obtained for links, which is related to the linking matrix.

Invariants of Rational Links Represented by Reduced Alternating Diagrams

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 26 August 2019Accepted: 02 June 2020Published online: 15 September 2020Keywordscontinued fractions, knots, links, braid index, alternating, Seifert graphAMS Subject HeadingsPrimary, 57M25; Secondary, 57M27Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Tree invariants and Milnor linking numbers with indeterminacy

This paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak, which are closely related to the classical Milnor linking numbers also known as [Formula: see text]-invariants. We prove that, analogously as for [Formula: see text]-invariants, certain residue classes of tree invariants yield link-homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor’s indeterminacy. For practical purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.

Ropelength, crossing number and finite-type invariants of links

Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of $n$-component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates and thus generalizing the known linking number bound. In the process, we collect several facts about finite type invariants and ropelength/crossing number of knots. We give examples of families of knots, where such estimates behave better than the well-known knot-genus estimate.

Milnor invariants via unipotent Magnus embeddings

We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of Milnor link and of several other links. Moreover, we refine the original Milnor invariants of higher degree.