Links In S3 Research Articles

THE GENUS OF PERIODIC LINKS WITH RATIONAL QUOTIENTS

AbstractIn this paper, we prove that if K is any periodic link in S3 whose quotient link is a 2-bridge link, then one half of the degree of the reduced Alexander polynomial, the minimal genus, the free genus and the canonical genus of K are all the same. We also give criteria to determine whether a given periodic link has a 2-bridge link quotient and some properties of this kind of periodic link.

Virtually fibered Montesinos links

We prove that all generalized Montesinos links in S3 which are not classic S L ˜ 2 -type are virtually fibred except the trivial link of two components. We also prove the virtually fibred property for a family of infinitely many classic Montesinos links of type S L ˜ 2 . As a byproduct we find the first family of infinitely many virtually fibred hyperbolic rational homology 3-spheres.

Invariants from classical field theory

We introduce a method that generates invariant functions from perturbative classical field theories depending on external parameters. By applying our methods to several field theories such as Abelian BF, Chern–Simons, and two-dimensional Yang–Mills theory, we obtain, respectively, the linking number for embedded submanifolds in compact varieties, the Gauss’ and the second Milnor’s invariant for links in S3, and invariants under area-preserving diffeomorphisms for configurations of immersed planar curves.

A FORMULA FOR THE DELTA-UNKNOTTING NUMBERS OF KNOTS

Recently, we introduced a description of knots and links in S3 in terms of integral matrices. The purpose of this paper is to show that the delta-unknotting numbers of the knots described by integral matrices with nonzero even entries coincide with the absolute values of the second coefficients of the Conway polynomials of the knots, and give an explicit formula for determining the delta-unknotting numbers of such knots.

A FACTORIZATION OF THE CONWAY POLYNOMIAL AND COVERING LINKAGE INVARIANTS

Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the [Formula: see text]-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ℤ[t, t-1]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t - 1 will be the linking numbers of certain derived links in S3. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S3. This generalizes a result of Hoste.

LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS

It is shown that Legendrian (respectively transverse) cable links in S3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J1(S1) with its standard tight contact structure.

Fox coloured knots and triangulations of $S^{3}$

A Fox coloured link is a pair (L,ω), where L is a link in S3 and ω a simple and transitive representation of π1(S3∖L) onto the symmetric group Σ3 on three elements. Here, a representation is called simple if it sends the meridians to transpositions. By works of the first two authors, any Fox coloured link (L,ω) gives rise to a closed orientable 3-manifold M(L,ω) equipped with a 3-fold simple covering p:M(L,ω)→S3 branched over L, and any closed orientable 3-manifold is homeomorphic to an M(K,ω) for some Fox coloured knot (K,ω) [see H. M. Hilden, Bull. Amer. Math. Soc. 80 (1974), 1243–1244; J. M. Montesinos, Bull. Amer. Math. Soc. 80 (1974), 845–846;]. In [Adv. Geom. 3 (2003), no. 2, 191–225;], I. V. Izmestʹev and M. Joswig proved that a triangulation of S3 gives rise in a natural way to some graph G on S3 and a representation of π1(S3∖G) into the symmetric group Σm for some m≤4. They also proved that any pair (L,ω), where L is a link in S3 and ω a simple (not necessarily transitive) representation of π1(S3∖L) into the symmetric group Σ4, can be obtained from a triangulation of S3. The proof that Izmestʹev and Joswig gave of this result is non-constructive. In the paper under review, the authors give a constructive proof of the same result. In particular, given a pair (L,ω) consisting of a link L in S3 and a simple (not necessarily transitive) representation of π1(S3∖L) onto the symmetric group Σ4, they construct a triangulation of S3 that gives rise to (L,ω) in a natural way.

REAL ANALYTIC GERMS $f \bar{g}$ AND OPEN-BOOK DECOMPOSITIONS OF THE 3-SPHERE

Let f,g: (C2,0) → (C,0) be two holomorphic germs with isolated singularities and no common branches and let Lf, [Formula: see text] be their links. We prove that the real analytic germ [Formula: see text] has an isolated singularity at 0 if and only if the link Lf ∪ -Lg is fibred. This was conjectured by Rudolph in [14]. If this condition holds, then the underlying Milnor fibration is an open-book fibration of the link Lf ∪ -Lg which coincides with [Formula: see text] in a tubular neighbourhood of this link. This enables one to realize a large family of fibrations of plumbing links in S3 as the Milnor fibrations of some real analytic germs [Formula: see text].

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

Stallings Twists Which Can be Realized by Plumbing and Deplumbing Hopf Bands

We define a type of Stallings twists, which represents a "complexity" of the twist, and show that the Stallings twists of a certain type on a fiber surface for a fibered link in S3 can be realized by plumbing one Hopf band and deplumbing another Hopf band. Using this technique, we construct stable Hopf plumbings which are not Hopf plumbings with an arbitrary high first Betti number.

Symmetry of Links and Classification of Lens Spaces

We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of 'symmetric' links in S3.

ORIENTED DIVIDES AND PLANE CURVE BRAIDS

This paper begins with a review of the extension of A'Campo's divides to the more general construction of oriented divides, using the same disk model of S3. It is shown that all links in S3 can be represented by oriented divides (in contrast with divide links) and simple methods are presented for moving between links and their oriented divide presentations. Using the properties of the disk model, oriented divides are extended by three extra equivalence rules, and it is shown that this extended theory of oriented divides is equivalent to knot theory in S3. Fueled by this result, the concept of oriented divide braid is introduced, together with a group construction which parallels the classical braid group representation of geometric braids. This leads to results analagous to Alexander's well-known theorem concerning braid presentations, and also the Markov theorem.

ON NON-SIMPLE REFLEXIVE LINKS

We consider Dehn surgeries on some non-simple 2-component links in S3 which yield S3. As corollary we give a class of 2-component links which are determined by their complements.

An algorithm producing a standard spine of a 3-manifold presented by surgery along a link

We provide a simple algorithm which produces a (branched) standard spine of a 3-manifold presented by surgery along a framed link inS3, giving an explicit upper bound on the complexity of the spine in terms of the complexity of a diagram of the link. As a corollary, we get an easy constructive proof of Casler’s result on the existence of a standard spine for a closed 3-manifold. We also describe an o-graph which represents the spine.

The fourth Skein Module and the Montesinos-Nakanishi Conjecture for 3-algebraic links

We study the fourth skein module of 3-manifolds, based on the skein relation b0L0 + b1L1 + b2L2 + b3L3 = 0 and a framing relation L(1) = aL (a, b0, b3 invertible). We give necessary conditions for trivial links to be linearly independent in the module. We show how elements of the skein module behave under the n-move and we compute the values for (2, n)-torus links and twist knots as elements of the skein module. Using mutants and rotors, we find different links which represent the same element in the skein module. We also show that algebraic links (in the sense of Conway) and closed 3-braids are linear combinations of trivial links. We introduce the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a by product we prove the Montesinos-Nakanishi 3-moves conjecture for 3-algebraic links (including 3-bridge links). For links in S3, the structure of our skein module suggests the existence of three new polynomial invariants of unoriented framed (or unframed) links. One of them would generalize the Kauffman polynomial of links and another one could be used to analyze amphicheirality of links (and may work better than the Kauffman polynomial). In the conclusion, we speculate that our new knot invariants are related to a deformation of the symplectic quotient of braid groups.

ON p-SUMMETRIC HEEGAARD SPLITTINGS

We show that every p-fold strictly-cyclic branched covering of a b-bridge link in S3 admits a p-symmetric Heegaard splitting – in the sense of Birman and Hilden – of genus g = (b - 1)(p - 1). This gives a complete converse of one of the results of the two authors. Moreover, we introduce the concept of weakly p-symmetric Heegaard splittings and prove similar results in this more general context.

REPRESENTATIVE BRAIDS FOR LINKS ASSOCIATED TO PLANE IMMERSED CURVES

In [ AC 2], A'Campo associates a link in S3 to any proper generic immersion of a disjoint union of arcs into a 2-disc. We give a sample algorithmic way to produce, from the immersion, a representative braid for such links. As a by-product we get a minimal representative braid for any algebraic link, from a divide associated to a real deformation of the polynomial defining the link.

A HOMFLY-PT POLYNOMIAL OF LINKS IN A SOLID TORUS

In this paper a Homfly-pt type Laurant polynomial1 for links in S1 × D2 is defined. This invariant is defined by giving an algebra and a trace, analogous to the Hecke algebra construction of the Homfly-pt polynomial for links in S3.

DISTINGUISHING LINK-HOMOTOPY CLASSES BY PRE-PERIPHERAL STRUCTURES

An open problem in link-homotopy of links in S3 is classification using peripheral invariants, analogous to that of Waldhausen for links up to ambient isotopy. An approach to such a classification was outlined by Levine, but shown not to be feasible by the author. Here, we develop an approach to finding classification counterexamples. The approach requires non-injectivity of a group homomorphism that is completely determined by minimal-weight commutator numbers (equivalent to the first non-vanishing [Formula: see text] invariants of Milnor). For non-injectivity, the minimal-weight commutator numbers must all be non-zero, and satisfy a certain system of polynomials, which we compute for 4- and 5-component links.

Handlebody Construction of Stein Surfaces

The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained-they correspond to open handlebodies with all handles of index < 2. An uncountable collection of exotic l4,'S is shown to admit Stein structures. New invariants of contact 3-manifolds are produced, including a complete (and computable) set of invariants for determining the homotopy class of a 2-plane field on a 3-manifold. These invariants are applicable to Seiberg-Witten theory. Several families of oriented 3-manifolds are examined, namely the Seifert fibered spaces and all surgeries on various links in S3, and in each case it is seen that most members of the family are the