Link Invariants Research Articles

Unlinking numbers of links with crossing number 10

In this paper we investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinking numbers of all but 2 of the 287 prime, non-split links with crossing number 10.

Logarithmic link invariants of [formula omitted] and asymptotic dimensions of singlet vertex algebras

We study relationships between the restricted unrolled quantum group U‾qH(sl2) at q=eπi/r, and the singlet vertex operator algebra M(r), r≥2. We use deformable families of modules to efficiently compute (1,1)-tangle invariants colored with projective U‾qH(sl2)-modules. These invariants relate to the colored Alexander tangle invariants studied in [6,40]. It follows that the regularized asymptotic dimensions of characters of M(r), studied previously by the first two authors, coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of M(r)-mod in connection to braided tensor categories.

Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. A state sum formula for the invariant is constructed via the chain-mail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the Crane-Yetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the Crane-Yetter invariant for nonmodular categories is stronger than signature and Euler invariant. A special case is the four-dimensional untwisted Dijkgraaf–Witten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in Walker-Wang models. Relations to different approaches to quantum gravity such as Cartan geometry and teleparallel gravity are also discussed.

The minimal coloring number of any non-splittable ℤ-colorable link is four

Ichihara and Matsudo introduced the notions of [Formula: see text]-colorable links and the minimal coloring number for [Formula: see text]-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable [Formula: see text]-colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable [Formula: see text]-colorable link is exactly 4.

On the skein theory of dichromatic links and invariants of finite type

In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in [Formula: see text]. They introduced a multi-variable polynomial invariant [Formula: see text]. We use special substitutions for some of the parameters of the invariant [Formula: see text] to show how to deduce invariants of finite type from [Formula: see text] using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in [Formula: see text] We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.

Skein Invariants of Links and Their State Sum Models

We present the new skein invariants of classical links, H [ H ] , K [ K ] and D [ D ] , based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. The invariants in this paper, were revealed through the skein theoretic definition of the invariants Θ d related to the Yokonuma–Hecke algebras and their 3-variable generalization Θ , which generalizes the Homflypt polynomial. H [ H ] is the regular isotopy counterpart of Θ . The invariants K [ K ] and D [ D ] are new generalizations of the Kauffman and the Dubrovnik polynomials. We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant Θ and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally.

Remarks on coloured triply graded link invariants

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.

Polynomial invariants for virtual marked graphs and virtual surface-link theory I

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

Classical link invariants from the framizations of the Iwahori–Hecke algebra and the Temperley–Lieb algebra of type A

In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma–Hecke algebras [Formula: see text], which are not topologically equivalent to the Homflypt polynomial. We then present the algebra [Formula: see text] which is the appropriate Temperley–Lieb analogue of [Formula: see text], as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma–Hecke algebra, and also its quotient, the partition Temperley–Lieb algebra [Formula: see text] and we prove an isomorphism of this algebra with a subalgebra of [Formula: see text].

Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1 dimensions are explored. Many of our TQFTs are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf–Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf–Witten description and all fermionic TQFTs (namely the spin TQFTs) are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We analytically calculate both the Abelian and non-Abelian braiding statistics data of anyonic particle and string excitations in these theories, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. Namely, we derive path integral expectation values of links formed by line and surface operators in these TQFTs. The acquired link invariants include not only the familiar Aharonov–Bohm linking number, but also Milnor triple linking number in 3 dimensions, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 4 dimensions. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(–Brown–Kervaire), Sato–Levine and others. We propose a new relation between the fermionic SPT partition function and the Rokhlin invariant. As an example, we can use these invariants and other physical observables, including ground state degeneracy, reduced modular Sxy and Txy matrices, and the partition function on RP3 manifold, to identify all ν∈Z8 classes of 2+1 dimensional gauged Z2-Ising-symmetric Z2f-fermionic Topological Superconductors (realized by stacking ν layers of a pair of chiral and anti-chiral p-wave superconductors [p+ip and p−ip], where boundary supports non-chiral Majorana–Weyl modes) with continuum spin-TQFTs.

Linking polynomials of virtual string links

In this paper, we define two polynomial invariants for virtual string links. One can be defined only on 2-component virtual string links. The other is defined on virtual string links with arbitrary components. Both invariants can distinguish virtual string links from their mirror images. In addition, we also show that these two invariants are link homotopic invariants. In the end, we give some interesting examples to show that the two invariants are different.

Generalizations of the Wada representations and virtual link groups

We extend the Wada representations of the classical braid group to the virtual and welded braid groups. Using the resulting representations, we construct the groups of virtual links and prove that they are link invariants. We give some examples of calculating the groups of torus (virtual) links.

On the Iwasawa invariants for links and Kida’s formula

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on [Formula: see text]-invariants in a [Formula: see text]-extension of [Formula: see text]-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of [Formula: see text]-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched [Formula: see text]-cover as an inverse system of cyclic branched [Formula: see text]-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

Multi-boundary entanglement in Chern-Simons theory and link invariants

We consider Chern-Simons theory for gauge group G at level k on 3-manifolds Mn with boundary consisting of n topologically linked tori. The Euclidean path integral on Mn defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where Mn is the link-complement of some n-component link inside the three-sphere S3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1)k) we give a general formula for the entanglement entropy associated to an arbitrary (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2)k, we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state).

Rational Whitney tower filtration of links

We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which are conjectured to be nontrivial, measure the potential difference between the Whitney tower theory in rational homology 4-balls and that in the 4-ball extensively developed by Conant, Schneiderman and Teichner.

Extended quandle spaces and shadow homotopy invariants of classical links

In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle.

Link invariant and $G_2$ web space

In this paper, we reconstruct Kuperberg’s $G_2$ web space [5, 6]. We introduce a new web diagram (a trivalent graph with only double edges) and new relations between Kuperberg’s web diagrams and the new web diagram. Using the web diagrams, we give crossing formulas for the $R$-matrices associated to some irreducible representations of $U_q(G_2)$ and calculate $G_2$ quantum link invariants for generalized twist links.

Representation of virtual braid groups to rook algebras and virtual links invariants

In 2011, Bigelow, Ramos and Yi constructed a representation of the group [Formula: see text] to subalgebra of planar diagrams [Formula: see text] of the rook algebra [Formula: see text]. We show that this representation of classical braid groups to the rook algebras can be extended to a representation of the virtual braid groups. This representation is used for constructing invariants of virtual links.

Knot invariants and M-theory: Hitchin equations, Chern-Simons actions, and surface operators

Recently Witten introduced a type IIB brane construction with certain boundary conditions to study knot invariants and Khovanov homology. The essential ingredients used in his work are the topologically twisted N = 4 Yang-Mills theory, localization equations and surface operators. In this paper we extend his construction in two possible ways. On one hand we show that a slight modification of Witten's brane construction could lead, using certain well defined duality transformations, to the model used by Ooguri-Vafa to study knot invariants using gravity duals. On the other hand, we argue that both these constructions, of Witten and of Ooguri-Vafa, lead to two different seven-dimensional manifolds in M-theory from where the topological theories may appear from certain twisting of the G-flux action. The non-abelian nature of the topological action may also be studied if we take the wrapped M2-brane states in the theory. We discuss explicit constructions of the seven-dimensional manifolds in M-theory, and show that both the localization equations and surface operators appear naturally from the Hamiltonian formalism of the theories. Knots and link invariants are then constructed using M2-brane states in both the models.

On the existence of real R-matrices for virtual link invariants

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.