Unknotted Components Research Articles

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.

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.

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 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.

A new basis for the Homflypt skein module of the solid torus

In this paper we give a new basis, Λ, for the Homflypt skein module of the solid torus, S(ST), which topologically is compatible with the handle sliding moves and which was predicted by J.H. Przytycki. The basis Λ is different from the basis Λ′, discovered independently by Hoste and Kidwell [1] and Turaev [2] with the use of diagrammatic methods, and also different from the basis of Morton and Aiston [3]. For finding the basis Λ we use the generalized Hecke algebra of type B, H1,n, which is generated by looping elements and braiding elements and which is isomorphic to the affine Hecke algebra of type A [4]. More precisely, we start with the well-known basis Λ′ of S(ST) and an appropriate linear basis Σn of the algebra H1,n. We then convert elements in Λ′ to sums of elements in Σn. Then, using conjugation and the stabilization moves, we convert these elements to sums of elements in Λ by managing gaps in the indices, by ordering the exponents of the looping elements and by eliminating braiding tails in the words. Further, we define total orderings on the sets Λ′ and Λ and, using these orderings, we relate the two sets via a block diagonal matrix, where each block is an infinite lower triangular matrix with invertible elements in the diagonal. Using this matrix we prove linear independence of the set Λ, thus Λ is a basis for S(ST).S(ST) plays an important role in the study of Homflypt skein modules of arbitrary c.c.o. 3-manifolds, since every c.c.o. 3-manifold can be obtained by integral surgery along a framed link in S3 with unknotted components. In particular, the new basis of S(ST) is appropriate for computing the Homflypt skein module of the lens spaces. In this paper we provide some basic algebraic tools for computing skein modules of c.c.o. 3-manifolds via algebraic means.

Satellite operators with distinct iterates in smooth concordance

Each pattern P P in a solid torus gives a function P : C → C P:\mathcal {C} \rightarrow \mathcal {C} on the smooth knot concordance group, taking any knot K K to its satellite P ( K ) P(K) . We give examples of winding number one patterns P P and a class of knots K K , such that the iterated satellites P i ( K ) P^i(K) are distinct in concordance, i.e. if i ≠ j ≥ 0 i \neq j \geq 0 , P i ( K ) ≠ P j ( K ) P^i(K) \neq P^j(K) . This implies that the operators P i P^i give distinct functions on C \mathcal {C} , providing further evidence for the (conjectured) fractal nature of C \mathcal {C} . Our theorem also allows us to construct several sets of examples, such as infinite families of topologically slice knots that are distinct in smooth concordance, infinite families of 2–component links (with unknotted components and linking number one) which are not smoothly concordant to the positive Hopf link, and infinitely many prime knots which have the same Alexander polynomial as an L L –space knot but are not themselves L L –space knots.

Nonconcordant links with homology cobordant zero-framed surgery manifolds

We use topological surgery in dimension four to give sufficient conditions for the zero framed surgery manifold of a 3-component link to be homology cobordant to the 3-torus, which arises from zero framed surgery on the Borromean rings, via a topological homology cobordism preserving the homotopy classes of the meridians. This enables us to give new examples of 3-component links with unknotted components and vanishing pairwise linking numbers, such that any two of these links have homology cobordant zero surgeries in the above sense, but the zero surgery manifolds are not homeomorphic. Moreover the links are not concordant to one another, and in fact they can be chosen to be height n but not height n+1 symmetric grope concordant, for each n which is at least three.

Concordance to links with unknotted components

We show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category which are topologically slice. We also give generalizations regarding components of prescribed Alexander polynomials. The main tools are covering link calculus, algebraic invariants of rational knot concordance theory, and the correction term of Heegaard Floer homology.

Smooth concordance of links topologically concordant to the Hopf link

It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim Davis. We construct infinitely many concordance classes of such links, and show that they have the stronger property of not being smoothly concordant to the Hopf link with knots tied in the components.

Brunnian links are determined by their complements

If L_1 and L_2 are two Brunnian links with all pairwise linking numbers 0, then we show that L_1 and L_2 are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If L_1 is a Brunnian link with all pairwise linking numbers 0, and the complement of L_2 is homeomorphic to the complement of L_1, then we show that L_2 may be obtained from L_1 by a sequence of twists around unknotted components. Finally, we show that for any positive integer n, an algorithm for detecting an n-component unlink leads immediately to an algorithm for detecting an unlink of any number of components. This algorithmic generalization is conceptually simple, but probably computationally impractical.

A Family of Links and the Conway Calculus

In 1969, J. H. Conway gave efficient methods of calculating abelian invariants of classical knots and links. The present paper includes a detailed exposition (with new proofs) of these methods and extensions in several directions. The main application given here is as follows. A link $L$ of two unknotted components in ${S^3}$ has the distinct lifting property for $p$ if the lifts of each component to the $p$-fold cover of ${S^3}$ branched along the other are distinct. The $p$-fold covers of these lifts are homeomorphic, and so $L$ gives an example of two distinct knots with the same $p$-fold cover. The above machinery is then used to construct an infinite family of links, each with the distinct lifting property for all $p \geqslant 2$.

A family of links and the Conway calculus

In 1969, J. H. Conway gave efficient methods of calculating abelian invariants of classical knots and links. The present paper includes a detailed exposition (with new proofs) of these methods and extensions in several directions. The main application given here is as follows. A link L L of two unknotted components in S 3 {S^3} has the distinct lifting property for p p if the lifts of each component to the p p -fold cover of S 3 {S^3} branched along the other are distinct. The p p -fold covers of these lifts are homeomorphic, and so L L gives an example of two distinct knots with the same p p -fold cover. The above machinery is then used to construct an infinite family of links, each with the distinct lifting property for all p ⩾ 2 p \geqslant 2 .