Slice Knots Research Articles

Infinite order amphicheiral knots

In answer to a question of Long, Flapan constructed an example of a prime strongly positive amphicheiral knot that is not slice. Long had proved that all such knots are algebraically slice. Here we show that the concordance group of algebraically slice knots contains an infinitely generated free subgroup that is generated by prime strongly positive amphicheiral knots. A simple corollary of this result is the existence of positive amphicheiral knots that are not of order two in concordance.

Knot concordance and torsion

Abstract Algebra, 2nd ed. , Prentice Hall, New Jersey, 1999.[F] R. Fox, A quick trip through knot theory , 1962 Topology of 3–manifolds and relatedtopics (Proc. The Univ. of Georgia Institute, 1961) 120–167 Prentice-Hall, EnglewoodCliffs, N.J.[FM] R. Fox and J. Milnor, Singularities of 2 –spheres in 4 –space and cobordism of knots ,Osaka J. Math. 3 (1966) 257–267.[G] C. Gordon, Problems , in Knot Theory, ed. J.-C. Hausmann, Springer Lecture Notesno. 685, 1977.[Gi] P. Gilmer, Slice knots in S 3 , Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135,305–322.[J] B. Jiang, A simple proof that the concordance group of algebraically slice knots isinfinitely generated , Proc. Amer. Math. Soc. 83 (1981), 189–192.[K1] R. Kirby, Problems in low dimensional manifold theory , in Algebraic and GeometricTopology (Stanford, 1976), vol 32, part II of Proc. Sympos. Pure Math., 273–312.[K2] R. Kirby, Problems in low dimensional manifold theory , Geometric Topology, AMS/IPStudies in Advanced Mathematics, ed. W. Kazez, 1997.[L1] J. Levine,

An obstruction to slicing knots using the eta invariant

We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.

On equivariant slice knots

We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knots— Naik’s and Choi-Ko-Song’s improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots — but are not equivariantly slice.

Note on the Casson-Gordon invariant of a satellite knot

In this paper we establish a relation between an appropriate version of the Casson-Gordon invariant of a satellite knot and those of its orbit and companion. We note that in some cases the contribution from, the companion falls. This gives a way to construct algebraically but not smoothly slice knots.

Linear independence of topologically slice knots in the smooth cobordism group

By using a result of M. Furuta concerning the homology cobordism group of homology 3-spheres, we give an infinite family of topologically slice knots which are linearly independent in the smooth knot-cobordism group.

Discriminants of Casson–Gordon invariants

Casson–Gordon invariants were first used to prove that certain algebraically slice knots in S3 are not slice knots [2, 3]. Since then they have been applied to a wide range of problems, including embedding problems and questions relating to boundary links [2, 10, 21, 25]. The most general Casson–Gordon invariant takes its value in L0(ℚ(ζd)(t)) ⊗ ℚ; here ζd denotes a primitive dth root of unity. Litherland [20] observed that one could usually tensor with ℤ(2) instead of ℚ, and in this way preserve the 2-torsion in the Witt group. He then constructed new examples of non-slice genus two knots which were detected with torsion classes in L0(ℚ(ζd)) ⊗ ℤ(2) modulo the image of L0(ℚ(ζd)) ⊗ ℤ(2).

On Slice Knots in the Complex Projective Planet

On Slice Knots in the Complex Projective Planet

Ribbon knot families via stallings' twists

AbstractWe investigate the effect on the Jones polynomial of a ribbon knot when two of its bands are twisted together. We use our results to prove that each of the three S-equivalence classes of genus 2 fibered doubly slice knots in S3 can be represented by infinitely many distinct prime fibered doubly slice ribbon knots.

Metabolic and hyperbolic forms from knot theory

In codimension two knot theory various signature invariants arise. We show that they all vanish on double slice knots but only certain linear combinations vanish on slice knots. This is done in the context of constructing complete sets of invariants of ± hermitian torsion forms over any Laurent polynomial ring F[ t, t −1] ( F a field) to detect metabolic or hyperbolic forms. Finally a precise formula relating two different versions of these signature invariants is given which shows that one is strictly weaker than the other.

Periodic ends and knot concordance

Some topological applications are given for Taubes' theorem on nonexistence of certain definite 4-manifolds with periodic ends. In particular, it is shown that the group of topologically slice knots up to smooth concordance is larger than Z , even modulo torsion. A corollary of the proof is that there are uncountably many diffeomorphism types of Casson handles. A specific collection of doubled knots is exhibited, no two members of which are smoothly concordant.

The Casson-Gordon invariants in high-dimensional knot theory

The Casson-Gordon invariants of knots in all dimensions are interpreted in terms of surgery theory. Applications are given to finding non-doubly slice knots, and doubly slice knots which are not the double of a disk knot. In even dimensions, the property of being doubly slice is shown to be largely homotopy theoretic, while in odd dimensions the surgery-theoretic method shows such properties to depend on more than the homotopy type.

Smooth concordance of topologically slice knots

Smooth concordance of topologically slice knots

The geometry of the slice-ribbon problem

One of the principal problems in low-dimensional topology concerns the question of whether all slice knots are ribbon(3),(4). It will be shown in this paper that this problem is closely related to geometric properties of discs properly embedded in the unit 4-ball in Euclidean space. The main result, Theorem 1·13, states that a knot is ribbon if and only if a representative within its isotopy class bounds an embedded minimal disc, and that this in turn happens if and only if a representative bounds a disc of sufficiently small total curvature.

On a Realization of Prime Tangles and Knots

The notion of a prime tangle is introduced by Kirby and Lickorish [7]. It is related deeply to the notion of a prime knot by the following result in [8]: summing together two prime tangles gives always a prime knot.The purpose of this paper is to exploit this above mentioned result of Lickorish in creating or detecting prime knots which satisfy certain properties. First, we shall express certain knots (two-bridge knots and Terasaka slice knots [14]) as a sum of a prime tangle and an untangle (the existence of such a sum is proven to every knot in [7] and is not unique) in a natural way (natural means here depending on certain specific geometrical characters of the class of knots). Second, every Alexander polynomial (or Conway polynomial) is shown to be realized by a prime algebraic knot (algebraic in the sense of Conway [3], Bonahon-Siebenmann [2]) which can be expressed as the sum of two prime algebraic tangles.

Twisting to algebraically slice knots

Twisting to algebraically slice knots

On the slice genus of knots

Given a knot K in the 3-sphere, the genus of K, denoted g(K), is defined to be the minimal genus for a Seifert surface for K. The slice genus gs(K) is defined to be the minimal genus of an oriented surface G admitting a smooth proper embedding in the 4-ball which maps CG to K. If we insist the embedding of G have no local maximum with respect to the radial function, we obtain the ribbon genus gr(K) instead. Thus a knot is slice (ribbon) if and only if gs(K) = 0 (gr(K)= 0). It is clear that g.~(K)<g,.(K)<=g(K). There are well known lower bounds on gs(K) given by invariants of a Seifert matrix for K. These are all included in the invariant re(K) [8] defined by Taylor. It gives the best possible bound based on a Seifert matrix, re(K) vanishes if and only if the Seifert pairing is metabolic. If this is the case, K is called algebraically slice. The work of Casson and Gordon [1, 2, 5] showed that certain algebraically slice knots are not in fact slice. We generalize the main theorem of [1]. As an application, we give a sequence of algebraically slice knots Q, such that gs(Q,)=g(Q,,)=n. We also study the slice genus of K, 41=K, where K, denotes the t twisted double of the unknot. We show for example that g.~(K 12 =~ K 1 z ) = 2 . K I2 is algebraically slice but not slice by [ l ] . Section 1 has some preliminaries on the linking form. In Sect. 2, we state and prove our main theorem. In Sect. 3 we give our examples. In this paper, all manifolds are oriented. We use e to denote the Euler characteristic.

A Simple Proof that the Concordance Group of Algebraically Slice Knots is Infinitely Generated

A Simple Proof that the Concordance Group of Algebraically Slice Knots is Infinitely Generated

A simple proof that the concordance group of algebraically slice knots is infinitely generated

A simple proof of the result stated in the title is obtained by making the Casson-Gordon invariant additive.

Trivializing ribbon links by Kirby moves

In this note it is shown that any ribbon link is a sublink of a ribbon link for which surgery on the longitudes gives a connected sum of copies of S1 x S2. In particular there are many links for which the analogue of the knot theoretic Property R fails, and sublinks of homology boundary links need not be homology boundary links. Higher dimensional analogues of these results are also given and it is shown that if n ≥ 2 the group of a ν-component ribbon n-link has a presentation of deficiency ν. Hence there are high dimensional slice knots which are not ribbon knots.