2-component Links Research Articles

H(2)-moves on torus links of type (2,2n)

We consider an [Formula: see text]-move between two torus links of type [Formula: see text], [Formula: see text]. We give some necessary conditions for two links [Formula: see text] and [Formula: see text] which are related by a single [Formula: see text]-move. In particular, we show that [Formula: see text] is obtained by a single [Formula: see text]-move from the Hopf link if and only if [Formula: see text], [Formula: see text], and [Formula: see text] is obtained by a single [Formula: see text]-move from the trivial 2-component link if and only if [Formula: see text], [Formula: see text],

Examples of topologically unknotted tori

We show that certain smooth tori with group Z \mathbb {Z} in S 4 S^4 have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all Z \mathbb {Z} -surfaces in S 4 S^4 are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of S 4 S^4 branched along any of these surfaces is a potentially exotic copy of S 2 × S 2 S^2 \times S^2 , and, in the case of turned twisted tori, we show they cannot be distinguished from S 2 × S 2 S^2 \times S^2 using Seiberg–Witten invariants.

Classification of 2-component virtual links up to ${\varXi }$-moves

Classification of 2-component virtual links up to ${\varXi }$-moves

Applications of the Casson-Walker invariant to the knot complement and the cosmetic crossing conjectures

We give a rational surgery formula for the Casson-Walker invariant of a 2-component link in $$S^{3}$$ which is a generalization of Matveev-Polyak’s formula. As application, we give more examples of a non-hyperbolic L-space M such that knots in M are determined by their complements. We also apply the result for the cosmetic crossing conjecture.

Topological Process of Splitting DNA-Links

A DNA replicon is modeled by a special type of 2-component link, called a DNA-link, in which two circles form a double helix around a trivial center core curve. The DNA replication process is semi-conservative, which is interpreted as a splitting process of the DNA-link. To split this non-trivial link, the linking number must become zero, and thus an unknotting operation is necessary. Some families of enzymes act as the unknotting operation. The present paper considers two topological problems; one is to know how the linking number is reduced and the other, how the enzymes are allocated at appropriate places. For the first problem, we suggest a reduction system of the linking number of a DNA-link. From this system, the number of repetitions of the procedure is obtained and this could be reduced when the DNA is previously relaxed by type I topoisomerases. For the second problem, we propose a possible conformation of the DNA-link in which the unknotting operation does not change the knot type of the core curve but decreases the writhe. This conformation could allocate type II topoisomerases to appropriate places. These models suggest that the combination of type I and type II topoisomerases efficiently reduces the linking number and it is possible to allocate enzymes by the conformation of DNA strands.

Doubly Slice Montesinos Links

In this paper, we compare the notions of double sliceness for links. The main result is showing that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end, we prove a result classifying Seifert fibered spaces that admit a smooth embedding into integer homology S1×S3s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links are also given.

Moves relating C-complexes: A correction to Cimasoni's “A geometric construction of the Conway potential function”

In work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms of a generalization of a Seifert surface for a link called a C-complex [2]. These surfaces were introduced by Cooper [9,10] for 2-component links and were used to compute Alexander invariants, signatures and nullities. Cooper presents a sequence of geometric moves which relate any two C-complexes for a 2-component link. Cimasoni gives an extension of Cooper's moves, providing a family of moves which relates any two C-complexes for a fixed link. This allows for an approach to studying links from the point of view of C-complexes and in following papers it has been used to derive invariants. Cimasoni's result is incorrect. We present counterexamples, a correction with detailed proof, and check that subsequent results relying on this lemma still hold.

L‐space surgeries on 2‐component L‐space links

In this paper, we analyze L-space surgeries on two component L-space links. We show that if one surgery coefficient is negative for the L-space surgery, then the corresponding link component is an unknot. If the link admits a very negative (that is, d 1 , d 2 ≪ 0 ) L-space surgery, it is either the unlink or the Hopf link. We also give a way to characterize the torus link T ( 2 , 2 l ) by observing an L-space surgery S d 1 , d 2 3 ( L ) with some d 1 d 2 < 0 on a 2-component L-space link with unknotted components. For some 2-component L-space links, we give explicit descriptions of the L-space surgery sets.

Pretzel links, mutation, and the slice-ribbon conjecture

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-stranded 2-component pretzel links.

A multivariable Casson–Lin type invariant

We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\operatorname{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable signatures. We also obtain some results concerning deformations of $\operatorname{SU}(2)$ representations of link groups.

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.

Writhe polynomials and shell moves for virtual knots and links

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a set of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented 2-component virtual links up to shell moves by using several invariants of virtual links.

2-Component links with genus two Heegaard splittings

In the present paper, we consider two types of 2-component links with genus two Heegaard splittings. One of them is an ordinary tunnel number one link, and the other is a somewhat different tunnel number one link. We will try to detect the differences between those two types. In fact, we will characterize composite tunnel number one links of the second type, and tunnel number one links of the second type with essential tori.

Coherent band-Gordian distances between knots and links with up to seven crossings

A coherent band surgery is a local move of an oriented knot or link by doing a surgery along an added band. The coherent band-Gordian distance between two links is the minimum number of coherent band surgeries needed to transform one into the other. We give a table of coherent band-Gordian distances between a knot and 2-component link with up to seven crossings.

Links with non-trivial Alexander polynomial which are topologically concordant to the Hopf link

We give infinitely many 2-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any 2-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.

A Symmetry Motivated Link Table

Proper identification of oriented knots and 2-component links requires a precise link nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible symmetry types for each link type. The study revisits the methods previously used to disambiguate chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two prime 2-component links with up to nine crossings. Guided by geometrical data, linking number, and the symmetry groups of 2-component links, canonical link diagrams for all but five link types (9 5 2, 9 34 2, 9 35 2, 9 39 2, and 9 41 2) are proposed. We include complete tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also prove a result on the behavior of the writhe under local lattice moves.

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.

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.

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).

A new family of links topologically, but not smoothly, concordant to the Hopf link

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so – in addition, they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously constructed by Cha–Kim–Ruberman–Strle; we show that our examples are distinct in smooth concordance from theirs.